Explanation:
Okay, so I need to find the measure of angle BAC in this diagram. Let me try to visualize it first. There's a triangle ABC with points A, B, and C. Point D is somewhere such that AD = BD = CD. Also, angle BCA is 40 degrees. The Asymptote code shows coordinates, but maybe I don't need to use coordinates. Let me think.

First, let me sketch the diagram mentally. Point D is connected to A, B, and C, and AD = BD = CD. So D is equidistant from A, B, and C. That suggests that D is the circumcenter of triangle ABC? Wait, the circumcenter is the intersection of the perpendicular bisectors and is equidistant from all three vertices. But if AD = BD = CD, then yes, D is the circumcenter. But wait, in that case, the circumradius would be AD = BD = CD. However, in a triangle, the circumcenter is located at the intersection of the perpendicular bisectors. But depending on the type of triangle, it can be inside or outside. But here, since D is connected to B and C, and the diagram shows D at the bottom between B and C? Wait, looking at the Asymptote code, coordinates are given: A is at (8.2635,9.8481), B is at (0,0), C is at (20,0), and D is at (10,0). So D is the midpoint of BC? Wait, BC is from (0,0) to (20,0), so midpoint would be at (10,0), which is D. But then AD = BD = CD? Let me check the coordinates. Let me compute AD, BD, CD.

First, coordinates:

A: (8.2635, 9.8481)

B: (0,0)

C: (20,0)

D: (10,0)

So BD is the distance from B(0,0) to D(10,0): that's 10 units.

CD is from C(20,0) to D(10,0): also 10 units. So BD = CD = 10. Now AD is from A(8.2635, 9.8481) to D(10,0). Let me compute that distance.

Difference in x: 10 - 8.2635 = 1.7365

Difference in y: 0 - 9.8481 = -9.8481

So distance AD is sqrt( (1.7365)^2 + (9.8481)^2 )

Let me compute that:

First, 1.7365 squared: approx (1.7365)^2 ≈ 3.015

Then 9.8481 squared: approx (9.8481)^2. Let me recall that 9.8481 is approximately 9.848, which is close to 10*sin(60°) or something? Wait, maybe exact value?

Wait, maybe these coordinates are chosen to make some angles exact? Let me check. The coordinates of A are (8.2635, 9.8481). Let me see if that corresponds to some known triangle.

Alternatively, maybe I can compute AD. Let me compute 1.7365 squared:

1.7365^2: Let me note that 1.732 is sqrt(3) ≈ 1.732, so 1.7365 is slightly larger. Let me compute 1.7365^2:

(1.732 + 0.0045)^2 ≈ (sqrt(3))^2 + 2*sqrt(3)*0.0045 + (0.0045)^2 ≈ 3 + 0.00819 + 0.00002 ≈ 3.0082. So approx 3.0082.

Then 9.8481 squared. Let me note that 9.8481 is approximately 9.848, which is 10*(sqrt(3)/something)? Wait, sqrt(3) is approx 1.732, so 9.848 is roughly 5.698*sqrt(3)? Wait, maybe not. Alternatively, maybe 9.848 is 20*sin(θ) or something?

Alternatively, maybe I can check if AD is equal to BD=10? Let me compute sqrt(3.0082 + (9.8481)^2). Let me compute 9.8481 squared:

Well, 9.8481 is approximately 9.848. Let me compute 9.848^2:

First, 10^2 = 100, subtract 2*10*0.152 + (0.152)^2? Wait, no. Wait, 9.848 is 10 - 0.152. So (10 - 0.152)^2 = 100 - 2*10*0.152 + 0.152^2 = 100 - 3.04 + 0.0231 ≈ 96.9831. So approx 96.98. Then adding the previous 3.0082 gives approx 96.98 + 3.0082 ≈ 99.9882. So sqrt(99.9882) ≈ 9.9994, which is approximately 10. So AD ≈ 10. Therefore, AD = BD = CD = 10. So D is the circumcenter? Wait, but in that case, if D is the circumcenter, then the circumradius is 10. But in triangle ABC, if D is the circumcenter, then DA = DB = DC. However, in the diagram, D is the midpoint of BC, so if D is the circumcenter, then triangle ABC must be such that the midpoint of BC is the circumcenter. Which would mean that triangle ABC is such that the circumcenter lies at the midpoint of BC. That would happen only if ABC is a right-angled triangle, right? Because in a right-angled triangle, the circumcenter is at the midpoint of the hypotenuse. So if D is the midpoint of BC and the circumcenter, then angle BAC must be a right angle? Wait, but angle BCA is given as 40 degrees. Wait, but if ABC is right-angled at A, then angle BAC would be 90 degrees, but angle BCA is 40 degrees. Wait, but then angle ABC would be 50 degrees. But maybe that's not the case here. Wait, but according to the coordinates, D is the midpoint of BC, and AD is equal to BD and CD. So if D is the circumcenter, then ABC is a right-angled triangle with right angle at A? Wait, but if ABC is right-angled at A, then BC would be the hypotenuse, and the circumcenter would be the midpoint of BC. So that would make sense. But then angle BCA is 40 degrees. So angle at C is 40 degrees, angle at A is 90 degrees, so angle at B would be 50 degrees. But then angle BAC is 90 degrees? But the problem is asking for angle BAC. Wait, but maybe my assumption that D is the circumcenter is wrong? Because in that case, angle BAC would be 90 degrees, but maybe the answer is different?

Wait, but according to the coordinates, AD is approximately 10, which is equal to BD and CD. So if D is the circumcenter, then ABC is right-angled at A. But then angle BCA is 40 degrees. Let me check with coordinates. Let me compute angle BCA. Point C is at (20,0), point B is at (0,0), point A is at (8.2635, 9.8481). So angle at C is angle BCA. Let me compute that angle. To compute angle at C, we can use coordinates. Let me compute vectors CB and CA.

Vector CB is from C to B: (0 - 20, 0 - 0) = (-20, 0)

Vector CA is from C to A: (8.2635 - 20, 9.8481 - 0) = (-11.7365, 9.8481)

Then angle between vectors CB and CA is angle BCA. Let me compute the angle between (-20, 0) and (-11.7365, 9.8481). The formula for the angle between two vectors u and v is:

cos(theta) = (u · v)/(|u| |v|)

Compute the dot product:

(-20)(-11.7365) + (0)(9.8481) = 234.73 + 0 = 234.73

|u| is |CB| = sqrt((-20)^2 + 0^2) = 20

|v| is |CA| = sqrt((-11.7365)^2 + (9.8481)^2). Let me compute that:

(-11.7365)^2 ≈ 137.73 (since 11.7365^2 ≈ (approx 11.7365)^2. Wait, earlier we saw that AD^2 ≈ 100, but here, CA^2 is (-11.7365)^2 + (9.8481)^2. Wait, but earlier, when we computed AD^2, we had (1.7365)^2 + (9.8481)^2 ≈ 3.0082 + 96.98 ≈ 99.988 ≈ 100. So that suggests that AD ≈ 10. But here, CA^2 is (-11.7365)^2 + (9.8481)^2. Let me compute that:

First, (-11.7365)^2: Let me note that 11.7365 is approximately 10 + 1.7365. Wait, but earlier, AD was from A to D(10,0), which had x difference of 1.7365. So 11.7365 is 10 + 1.7365? Wait, no. Wait, the x-coordinate of A is 8.2635, so from C(20,0) to A(8.2635, 9.8481), the x difference is 8.2635 - 20 = -11.7365. So yes, that's correct. So (-11.7365)^2 is (approx) (11.7365)^2. Let me compute that. Let me note that 11.7365 is approximately 11.7365. Let me think of 11.7365 as 10 + 1.7365. Then (10 + 1.7365)^2 = 100 + 2*10*1.7365 + (1.7365)^2 ≈ 100 + 34.73 + 3.015 ≈ 137.745. Then adding (9.8481)^2 ≈ 96.98 gives total CA^2 ≈ 137.745 + 96.98 ≈ 234.725. Therefore, |CA| ≈ sqrt(234.725) ≈ 15.32.

So then cos(theta) = 234.73 / (20 * 15.32) ≈ 234.73 / 306.4 ≈ 0.766. Then arccos(0.766) is approximately 40 degrees. Because cos(40°) ≈ 0.766. Yes! So angle BCA is indeed 40 degrees, as given. Therefore, the coordinates are consistent with angle BCA being 40 degrees, and D being the midpoint of BC, with AD = BD = CD = 10. Therefore, triangle ABC is right-angled at A? Wait, if D is the circumcenter, then ABC is right-angled at A. But angle BAC is the right angle? Wait, but if ABC is right-angled at A, then angle BAC is 90 degrees. But the problem is asking for angle BAC. However, according to the coordinates, angle BAC is the angle at A between BA and CA. Let me compute that angle. Let me compute vectors AB and AC.

Point A is (8.2635, 9.8481), B is (0,0), C is (20,0). So vector AB is B - A = (0 - 8.2635, 0 - 9.8481) = (-8.2635, -9.8481). Vector AC is C - A = (20 - 8.2635, 0 - 9.8481) = (11.7365, -9.8481). Then angle between AB and AC is angle BAC. Let me compute that angle.

Again, using the dot product formula. Let me compute the dot product of AB and AC:

(-8.2635)(11.7365) + (-9.8481)(-9.8481)

First term: -8.2635 * 11.7365. Let me approximate this. Let me note that 8.2635 * 11.7365. Let me think of 8.2635 as approx 8.26 and 11.7365 as approx 11.74. Multiplying 8.26 * 11.74. Let me compute 8 * 11.74 = 93.92, 0.26 * 11.74 ≈ 3.0524, so total approx 93.92 + 3.0524 ≈ 96.97. So with the negative sign, it's -96.97.

Second term: (-9.8481)(-9.8481) = (9.8481)^2 ≈ 96.98 as before.

So total dot product ≈ -96.97 + 96.98 ≈ 0.01. Which is approximately zero. Therefore, the angle between AB and AC is approximately 90 degrees. Therefore, angle BAC is 90 degrees. But the problem says angle BCA is 40 degrees, and asks for angle BAC. If angle BAC is 90 degrees, then that would be the answer. But wait, the problem didn't specify that ABC is right-angled. However, according to the coordinates, it is. But maybe the problem is designed so that even without coordinates, we can figure it out?

Wait, but maybe I made a wrong assumption? Because if D is the circumcenter, then ABC is right-angled at A. But maybe D is not the circumcenter? Wait, but AD = BD = CD, so D is equidistant from A, B, C, so it must be the circumcenter. Therefore, ABC is right-angled at A. Therefore, angle BAC is 90 degrees. But then why is angle BCA given as 40 degrees? Because in a right-angled triangle, the other angles would add up to 90 degrees. So angle at C is 40 degrees, angle at B is 50 degrees, angle at A is 90 degrees. Therefore, angle BAC is 90 degrees. But the problem is presented with an Asymptote diagram, and maybe the answer is different? Wait, but according to coordinates, it is 90 degrees. However, maybe I need to verify again.

Wait, but maybe my calculation of the dot product was approximate. Let me check more accurately. The dot product of AB and AC was approximately zero. Let me compute more precisely.

First, AB vector is (-8.2635, -9.8481)

AC vector is (11.7365, -9.8481)

Dot product = (-8.2635)(11.7365) + (-9.8481)(-9.8481)

Let me compute (-8.2635)(11.7365):

Let me note that 8.2635 * 11.7365. Let me check if these numbers are related. Wait, 8.2635 is approximately 10 - 1.7365. Wait, 10 - 1.7365 = 8.2635. And 11.7365 is 10 + 1.7365. Therefore, multiplying (10 - 1.7365)(10 + 1.7365) = 10^2 - (1.7365)^2 = 100 - 3.015 ≈ 96.985. Therefore, (-8.2635)(11.7365) = - (10 - 1.7365)(10 + 1.7365) ≈ -96.985. Then the second term is (-9.8481)(-9.8481) = (9.8481)^2 ≈ 96.98. Therefore, total dot product ≈ -96.985 + 96.98 ≈ -0.005. Which is approximately zero. Therefore, the angle between AB and AC is approximately 90 degrees. Therefore, angle BAC is 90 degrees. Therefore, the answer is 90 degrees? But the problem is presented in a way that might not be expecting that? Wait, but according to the coordinates and calculations, it is 90 degrees. However, maybe there's a mistake here?

Wait, let me think again. If D is the circumcenter, then angle BAC is 90 degrees. But maybe D is not the circumcenter? Wait, but AD = BD = CD, so D is equidistant from A, B, C, so it must be the circumcenter. Therefore, triangle ABC must be right-angled at A. Therefore, angle BAC is 90 degrees. However, the problem says angle BCA is 40 degrees. So angle at C is 40 degrees, angle at A is 90 degrees, so angle at B is 50 degrees. Therefore, angle BAC is 90 degrees. Therefore, the answer is 90 degrees? But maybe the problem is designed to trick us? Wait, but according to the coordinates, it is 90 degrees. However, maybe I need to solve it without coordinates?

Let me try to solve it using pure geometry. Given that AD = BD = CD, so D is the circumcenter. Therefore, triangle ABC has circumcenter at D. Therefore, if D is the circumcenter, then the position of D depends on the triangle. If D is on BC, then BC must be the diameter of the circumcircle. Therefore, angle BAC is 90 degrees. Because in a circle, if BC is the diameter, then any point A on the circle forms a right angle with B and C. Therefore, angle BAC is 90 degrees. Therefore, regardless of the other angles, if D is the midpoint of BC and the circumcenter, then ABC is right-angled at A. Therefore, angle BAC is 90 degrees. Therefore, the answer is 90 degrees. But the problem says angle BCA is 40 degrees. So angle at C is 40 degrees, angle at A is 90 degrees, angle at B is 50 degrees. Therefore, angle BAC is 90 degrees. Therefore, the answer is 90 degrees. But maybe the problem is not in the way I think?

Wait, but maybe D is not the midpoint of BC? Wait, according to the Asymptote code, D is at (10,0), which is the midpoint of BC since B is at (0,0) and C is at (20,0). Therefore, BD = DC = 10. And AD is also 10. Therefore, D is the circumcenter. Therefore, ABC is right-angled at A. Therefore, angle BAC is 90 degrees. Therefore, answer is 90 degrees. But maybe the problem is from an Olympiad or something, and the answer is different? Wait, but according to the coordinates and calculations, it's 90 degrees. Maybe I need to check again.

Alternatively, maybe there is a different configuration? Let me think. Suppose that D is not on BC. Wait, but according to the Asymptote code, D is at (10,0), which is on BC. So BD = DC, and AD = BD. Therefore, D is the circumcenter. Therefore, ABC is right-angled at A. Therefore, angle BAC is 90 degrees. Therefore, answer is 90 degrees. But maybe the problem is in another configuration? Wait, maybe D is not on BC? Wait, the Asymptote code draws the triangle ABC with A connected to B, C, and D is connected to B, C, and A. Wait, the Asymptote code draws:

draw((0,0)--(10,0)--(8.2635,9.8481)--cycle,black+linewidth(1));

So that's triangle ABD? Wait, no. Wait, the first draw command is (0,0)--(10,0)--(8.2635,9.8481)--cycle, which connects B(0,0) to D(10,0) to A(8.2635,9.8481) and back to B? Wait, no, the cycle would connect back to the first point, so B to D to A to B? Then the second draw command is (10,0)--(20,0)--(8.2635,9.8481), which connects D(10,0) to C(20,0) to A(8.2635,9.8481) and back to D? So that's triangle ADC? Then there are some other lines. So overall, the diagram has triangle ABC with D on BC, and AD, BD, CD all equal. Therefore, D is midpoint of BC and circumradius. Therefore, ABC is right-angled at A. Therefore, angle BAC is 90 degrees. Therefore, answer is 90 degrees. But maybe the problem is in another configuration? Wait, but according to the Asymptote code, coordinates are given, so maybe the answer is 90 degrees. However, maybe I need to check with another method.

Alternatively, let me try to solve it without coordinates. Let me assume that AD = BD = CD. So D is the circumcenter. Therefore, angle BAC is equal to half the measure of the arc BC. Wait, no. In a circle, the central angle is twice the inscribed angle. Wait, if D is the circumcenter, then angle BDC is twice angle BAC. Wait, but since D is the circumcenter, BD and CD are radii. Wait, but if D is on BC, then BD = DC = AD. Therefore, triangle DBC is isoceles with BD = DC. Wait, but if D is on BC, then BD = DC implies that D is the midpoint of BC. Therefore, BC is the diameter of the circumcircle. Therefore, angle BAC is 90 degrees. Therefore, answer is 90 degrees. Therefore, regardless of other angles, angle BAC is 90 degrees. Therefore, answer is 90 degrees. Therefore, maybe the answer is 90 degrees. But why is angle BCA given as 40 degrees? Because in that case, angle at C is 40 degrees, angle at A is 90 degrees, angle at B is 50 degrees. Therefore, that's consistent. Therefore, answer is 90 degrees. Therefore, I think the answer is 90 degrees. But maybe I need to check again.

Wait, but maybe there's a mistake here. Suppose that D is not on BC? Wait, but according to the Asymptote code, D is on BC. But maybe in the problem, D is not on BC? Wait, the problem says "In the diagram, AD=BD=CD and angle BCA = 40 degrees." So the diagram is given, but since I don't have the diagram, but the Asymptote code shows D on BC. Therefore, assuming that, then answer is 90 degrees. However, maybe the problem is from an official source where the answer is different? Wait, but according to calculations, it's 90 degrees. Alternatively, maybe I misread the problem? Let me check again.

Wait, the problem says "AD=BD=CD". So D is equidistant from A, B, and C. Therefore, D is the circumcenter. Therefore, if D is the circumcenter, then ABC is a triangle with circumcenter D. If D is on BC, then BC is the diameter. Therefore, angle BAC is 90 degrees. Therefore, answer is 90 degrees. Therefore, I think that is correct. Therefore, maybe the answer is 90 degrees. However, maybe the problem is in three dimensions? No, it's a plane diagram. Therefore, I think the answer is 90 degrees. Therefore, I think the measure of angle BAC is 90 degrees. Therefore, boxed answer is \boxed{90}.

Wait, but let me check once more. Suppose angle BAC is 90 degrees, angle BCA is 40 degrees, then angle ABC is 50 degrees. Then, in triangle ABC, with BC as hypotenuse, then BD = DC = AD = R, the circumradius. In a right-angled triangle, the circumradius is half the hypotenuse. Therefore, BC = 2R. Therefore, BD = DC = R, which is half of BC. Therefore, D is the midpoint of BC. Therefore, yes, that's consistent. Therefore, all conditions are satisfied. Therefore, angle BAC is 90 degrees. Therefore, answer is 90 degrees. Therefore, I think that is correct.

**Final Answer**
\boxed{90}

To determine the measure of $\angle BAC$, we are given the following:

- $AD = BD = CD$
- $\angle BCA = 40^\circ$

---

### Step 1: Understanding the Geometry

The key condition is that $AD = BD = CD$. This implies that point $D$ is **equidistant** from points $A$, $B$, and $C$. In a triangle, the only point that is equidistant from all three vertices is the **circumcenter**. Therefore, point $D$ is the **circumcenter** of triangle $ABC$.

---

### Step 2: Properties of the Circumcenter

In any triangle, the circumcenter is the intersection of the perpendicular bisectors of the sides. It is also the center of the **circumcircle** of the triangle, and all three vertices lie on the circle centered at the circumcenter.

If the circumcenter lies on a side of the triangle (as in this case, since $D$ lies on segment $BC$), then that side must be the **diameter** of the circumcircle. This is a key geometric property: if a triangle is inscribed in a circle and one of its sides is the diameter, then the angle opposite that side is a **right angle**.

---

### Step 3: Applying the Right Angle Theorem

Since $D$ lies on segment $BC$ and is equidistant from $A$, $B$, and $C$, it must be the midpoint of $BC$, and thus $BC$ is the **diameter** of the circumcircle. Therefore, triangle $ABC$ is a **right triangle** with the right angle at point $A$.

This means:

$$
\angle BAC = 90^\circ
$$

---

### Step 4: Verifying with Given Angle

We are told that $\angle BCA = 40^\circ$. In a right triangle, the sum of the two non-right angles is $90^\circ$. Therefore, the remaining angle $\angle ABC = 50^\circ$, which is consistent with the triangle being right-angled at $A$.

---

### Final Conclusion

All conditions are satisfied with $\angle BAC = 90^\circ$, and the diagram confirms that $D$ is the midpoint of $BC$, making $BC$ the diameter of the circumcircle and $\angle BAC$ a right angle.

$$
\boxed{90}
$$

[Assistant]
Explanation:
Okay, so I need to find the measure of angle BAC in this diagram. Let me try to visualize it first. There's a triangle ABC with points A, B, and C. Point D is somewhere such that AD = BD = CD. Also, angle BCA is 40 degrees. The Asymptote code shows coordinates, but maybe I don't need to use coordinates. Let me think.

First, let me sketch the diagram mentally. Point D is connected to A, B, and C, and AD = BD = CD. So D is equidistant from A, B, and C. That suggests that D is the circumcenter of triangle ABC? Wait, the circumcenter is the intersection of the perpendicular bisectors and is equidistant from all three vertices. But if AD = BD = CD, then yes, D is the circumcenter. But wait, in that case, the circumradius would be AD = BD = CD. However, in a triangle, the circumcenter is located at the intersection of the perpendicular bisectors. But depending on the type of triangle, it can be inside or outside. But here, since D is connected to B and C, and the diagram shows D at the bottom between B and C? Wait, looking at the Asymptote code, coordinates are given: A is at (8.2635,9.8481), B is at (0,0), C is at (20,0), and D is at (10,0). So D is the midpoint of BC? Wait, BC is from (0,0) to (20,0), so midpoint would be at (10,0), which is D. But then AD = BD = CD? Let me check the coordinates. Let me compute AD, BD, CD.

First, coordinates:

A: (8.2635, 9.8481)

B: (0,0)

C: (20,0)

D: (10,0)

So BD is the distance from B(0,0) to D(10,0): that's 10 units.

CD is from C(20,0) to D(10,0): also 10 units. So BD = CD = 10. Now AD is from A(8.2635, 9.8481) to D(10,0). Let me compute that distance.

Difference in x: 10 - 8.2635 = 1.7365

Difference in y: 0 - 9.8481 = -9.8481

So distance AD is sqrt( (1.7365)^2 + (9.8481)^2 )

Let me compute that:

First, 1.7365 squared: approx (1.7365)^2 ≈ 3.015

Then 9.8481 squared: approx (9.8481)^2. Let me recall that 9.8481 is approximately 9.848, which is close to 10*sin(60°) or something? Wait, maybe exact value?

Wait, maybe these coordinates are chosen to make some angles exact? Let me check. The coordinates of A are (8.2635, 9.8481). Let me see if that corresponds to some known triangle.

Alternatively, maybe I can compute AD. Let me compute 1.7365 squared:

1.7365^2: Let me note that 1.732 is sqrt(3) ≈ 1.732, so 1.7365 is slightly larger. Let me compute 1.7365^2:

(1.732 + 0.0045)^2 ≈ (sqrt(3))^2 + 2*sqrt(3)*0.0045 + (0.0045)^2 ≈ 3 + 0.00819 + 0.00002 ≈ 3.0082. So approx 3.0082.

Then 9.8481 squared. Let me note that 9.8481 is approximately 9.848, which is 10*(sqrt(3)/something)? Wait, sqrt(3) is approx 1.732, so 9.848 is roughly 5.698*sqrt(3)? Wait, maybe not. Alternatively, maybe 9.848 is 20*sin(θ) or something?

Alternatively, maybe I can check if AD is equal to BD=10? Let me compute sqrt(3.0082 + (9.8481)^2). Let me compute 9.8481 squared:

Well, 9.8481 is approximately 9.848. Let me compute 9.848^2:

First, 10^2 = 100, subtract 2*10*0.152 + (0.152)^2? Wait, no. Wait, 9.848 is 10 - 0.152. So (10 - 0.152)^2 = 100 - 2*10*0.152 + 0.152^2 = 100 - 3.04 + 0.0231 ≈ 96.9831. So approx 96.98. Then adding the previous 3.0082 gives approx 96.98 + 3.0082 ≈ 99.9882. So sqrt(99.9882) ≈ 9.9994, which is approximately 10. So AD ≈ 10. Therefore, AD = BD = CD = 10. So D is the circumcenter? Wait, but in that case, if D is the circumcenter, then the circumradius is 10. But in triangle ABC, if D is the circumcenter, then DA = DB = DC. However, in the diagram, D is the midpoint of BC, so if D is the circumcenter, then triangle ABC must be such that the midpoint of BC is the circumcenter. Which would mean that triangle ABC is such that the circumcenter lies at the midpoint of BC. That would happen only if ABC is a right-angled triangle, right? Because in a right-angled triangle, the circumcenter is at the midpoint of the hypotenuse. So if D is the midpoint of BC and the circumcenter, then angle BAC must be a right angle? Wait, but angle BCA is given as 40 degrees. Wait, but if ABC is right-angled at A, then angle BAC would be 90 degrees, but angle BCA is 40 degrees. Wait, but then angle ABC would be 50 degrees. But maybe that's not the case here. Wait, but according to the coordinates, D is the midpoint of BC, and AD is equal to BD and CD. So if D is the circumcenter, then ABC is a right-angled triangle with right angle at A? Wait, but if ABC is right-angled at A, then BC would be the hypotenuse, and the circumcenter would be the midpoint of BC. So that would make sense. But then angle BCA is 40 degrees. So angle at C is 40 degrees, angle at A is 90 degrees, so angle at B would be 50 degrees. But then angle BAC is 90 degrees? But the problem is asking for angle BAC. Wait, but maybe my assumption that D is the circumcenter is wrong? Because in that case, angle BAC would be 90 degrees, but maybe the answer is different?

Wait, but according to the coordinates, AD is approximately 10, which is equal to BD and CD. So if D is the circumcenter, then ABC is right-angled at A. But then angle BCA is 40 degrees. Let me check with coordinates. Let me compute angle BCA. Point C is at (20,0), point B is at (0,0), point A is at (8.2635, 9.8481). So angle at C is angle BCA. Let me compute that angle. To compute angle at C, we can use coordinates. Let me compute vectors CB and CA.

Vector CB is from C to B: (0 - 20, 0 - 0) = (-20, 0)

Vector CA is from C to A: (8.2635 - 20, 9.8481 - 0) = (-11.7365, 9.8481)

Then angle between vectors CB and CA is angle BCA. Let me compute the angle between (-20, 0) and (-11.7365, 9.8481). The formula for the angle between two vectors u and v is:

cos(theta) = (u · v)/(|u| |v|)

Compute the dot product:

(-20)(-11.7365) + (0)(9.8481) = 234.73 + 0 = 234.73

|u| is |CB| = sqrt((-20)^2 + 0^2) = 20

|v| is |CA| = sqrt((-11.7365)^2 + (9.8481)^2). Let me compute that:

(-11.7365)^2 ≈ 137.73 (since 11.7365^2 ≈ (approx 11.7365)^2. Wait, earlier we saw that AD^2 ≈ 100, but here, CA^2 is (-11.7365)^2 + (9.8481)^2. Wait, but earlier, when we computed AD^2, we had (1.7365)^2 + (9.8481)^2 ≈ 3.0082 + 96.98 ≈ 99.988 ≈ 100. So that suggests that AD ≈ 10. But here, CA^2 is (-11.7365)^2 + (9.8481)^2. Let me compute that:

First, (-11.7365)^2: Let me note that 11.7365 is approximately 10 + 1.7365. Wait, but earlier, AD was from A to D(10,0), which had x difference of 1.7365. So 11.7365 is 10 + 1.7365? Wait, no. Wait, the x-coordinate of A is 8.2635, so from C(20,0) to A(8.2635, 9.8481), the x difference is 8.2635 - 20 = -11.7365. So yes, that's correct. So (-11.7365)^2 is (approx) (11.7365)^2. Let me compute that. Let me note that 11.7365 is approximately 11.7365. Let me think of 11.7365 as 10 + 1.7365. Then (10 + 1.7365)^2 = 100 + 2*10*1.7365 + (1.7365)^2 ≈ 100 + 34.73 + 3.015 ≈ 137.745. Then adding (9.8481)^2 ≈ 96.98 gives total CA^2 ≈ 137.745 + 96.98 ≈ 234.725. Therefore, |CA| ≈ sqrt(234.725) ≈ 15.32.

So then cos(theta) = 234.73 / (20 * 15.32) ≈ 234.73 / 306.4 ≈ 0.766. Then arccos(0.766) is approximately 40 degrees. Because cos(40°) ≈ 0.766. Yes! So angle BCA is indeed 40 degrees, as given. Therefore, the coordinates are consistent with angle BCA being 40 degrees, and D being the midpoint of BC, with AD = BD = CD = 10. Therefore, triangle ABC is right-angled at A? Wait, if D is the circumcenter, then ABC is right-angled at A. But angle BAC is the right angle? Wait, but if ABC is right-angled at A, then angle BAC is 90 degrees. But the problem is asking for angle BAC. However, according to the coordinates, angle BAC is the angle at A between BA and CA. Let me compute that angle. Let me compute vectors AB and AC.

Point A is (8.2635, 9.8481), B is (0,0), C is (20,0). So vector AB is B - A = (0 - 8.2635, 0 - 9.8481) = (-8.2635, -9.8481). Vector AC is C - A = (20 - 8.2635, 0 - 9.8481) = (11.7365, -9.8481). Then angle between AB and AC is angle BAC. Let me compute that angle.

Again, using the dot product formula. Let me compute the dot product of AB and AC:

(-8.2635)(11.7365) + (-9.8481)(-9.8481)

First term: -8.2635 * 11.7365. Let me approximate this. Let me note that 8.2635 * 11.7365. Let me think of 8.2635 as approx 8.26 and 11.7365 as approx 11.74. Multiplying 8.26 * 11.74. Let me compute 8 * 11.74 = 93.92, 0.26 * 11.74 ≈ 3.0524, so total approx 93.92 + 3.0524 ≈ 96.97. So with the negative sign, it's -96.97.

Second term: (-9.8481)(-9.8481) = (9.8481)^2 ≈ 96.98 as before.

So total dot product ≈ -96.97 + 96.98 ≈ 0.01. Which is approximately zero. Therefore, the angle between AB and AC is approximately 90 degrees. Therefore, angle BAC is 90 degrees. But the problem says angle BCA is 40 degrees, and asks for angle BAC. If angle BAC is 90 degrees, then that would be the answer. But wait, the problem didn't specify that ABC is right-angled. However, according to the coordinates, it is. But maybe the problem is designed so that even without coordinates, we can figure it out?

Wait, but maybe I made a wrong assumption? Because if D is the circumcenter, then ABC is right-angled at A. But maybe D is not the circumcenter? Wait, but AD = BD = CD, so D is equidistant from A, B, C, so it must be the circumcenter. Therefore, ABC is right-angled at A. Therefore, angle BAC is 90 degrees. But then why is angle BCA given as 40 degrees? Because in a right-angled triangle, the other angles would add up to 90 degrees. So angle at C is 40 degrees, angle at B is 50 degrees, angle at A is 90 degrees. Therefore, angle BAC is 90 degrees. But the problem is presented with an Asymptote diagram, and maybe the answer is different? Wait, but according to coordinates, it is 90 degrees. However, maybe I need to verify again.

Wait, but maybe my calculation of the dot product was approximate. Let me check more accurately. The dot product of AB and AC was approximately zero. Let me compute more precisely.

First, AB vector is (-8.2635, -9.8481)

AC vector is (11.7365, -9.8481)

Dot product = (-8.2635)(11.7365) + (-9.8481)(-9.8481)

Let me compute (-8.2635)(11.7365):

Let me note that 8.2635 * 11.7365. Let me check if these numbers are related. Wait, 8.2635 is approximately 10 - 1.7365. Wait, 10 - 1.7365 = 8.2635. And 11.7365 is 10 + 1.7365. Therefore, multiplying (10 - 1.7365)(10 + 1.7365) = 10^2 - (1.7365)^2 = 100 - 3.015 ≈ 96.985. Therefore, (-8.2635)(11.7365) = - (10 - 1.7365)(10 + 1.7365) ≈ -96.985. Then the second term is (-9.8481)(-9.8481) = (9.8481)^2 ≈ 96.98. Therefore, total dot product ≈ -96.985 + 96.98 ≈ -0.005. Which is approximately zero. Therefore, the angle between AB and AC is approximately 90 degrees. Therefore, angle BAC is 90 degrees. Therefore, the answer is 90 degrees? But the problem is presented in a way that might not be expecting that? Wait, but according to the coordinates and calculations, it is 90 degrees. However, maybe there's a mistake here?

Wait, let me think again. If D is the circumcenter, then angle BAC is 90 degrees. But maybe D is not the circumcenter? Wait, but AD = BD = CD, so D is equidistant from A, B, C, so it must be the circumcenter. Therefore, triangle ABC must be right-angled at A. Therefore, angle BAC is 90 degrees. However, the problem says angle BCA is 40 degrees. So angle at C is 40 degrees, angle at A is 90 degrees, so angle at B is 50 degrees. Therefore, angle BAC is 90 degrees. Therefore, the answer is 90 degrees? But maybe the problem is designed to trick us? Wait, but according to the coordinates, it is 90 degrees. However, maybe I need to solve it without coordinates?

Let me try to solve it using pure geometry. Given that AD = BD = CD, so D is the circumcenter. Therefore, triangle ABC has circumcenter at D. Therefore, if D is the circumcenter, then the position of D depends on the triangle. If D is on BC, then BC must be the diameter of the circumcircle. Therefore, angle BAC is 90 degrees. Because in a circle, if BC is the diameter, then any point A on the circle forms a right angle with B and C. Therefore, angle BAC is 90 degrees. Therefore, regardless of the other angles, if D is the midpoint of BC and the circumcenter, then ABC is right-angled at A. Therefore, angle BAC is 90 degrees. Therefore, the answer is 90 degrees. But the problem says angle BCA is 40 degrees. So angle at C is 40 degrees, angle at A is 90 degrees, angle at B is 50 degrees. Therefore, angle BAC is 90 degrees. Therefore, the answer is 90 degrees. But maybe the problem is not in the way I think?

Wait, but maybe D is not the midpoint of BC? Wait, according to the Asymptote code, D is at (10,0), which is the midpoint of BC since B is at (0,0) and C is at (20,0). Therefore, BD = DC = 10. And AD is also 10. Therefore, D is the circumcenter. Therefore, ABC is right-angled at A. Therefore, angle BAC is 90 degrees. Therefore, answer is 90 degrees. But maybe the problem is from an Olympiad or something, and the answer is different? Wait, but according to the coordinates and calculations, it's 90 degrees. Maybe I need to check again.

Alternatively, maybe there is a different configuration? Let me think. Suppose that D is not on BC. Wait, but according to the Asymptote code, D is at (10,0), which is on BC. So BD = DC, and AD = BD. Therefore, D is the circumcenter. Therefore, ABC is right-angled at A. Therefore, angle BAC is 90 degrees. Therefore, answer is 90 degrees. But maybe the problem is in another configuration? Wait, maybe D is not on BC? Wait, the Asymptote code draws the triangle ABC with A connected to B, C, and D is connected to B, C, and A. Wait, the Asymptote code draws:

draw((0,0)--(10,0)--(8.2635,9.8481)--cycle,black+linewidth(1));

So that's triangle ABD? Wait, no. Wait, the first draw command is (0,0)--(10,0)--(8.2635,9.8481)--cycle, which connects B(0,0) to D(10,0) to A(8.2635,9.8481) and back to B? Wait, no, the cycle would connect back to the first point, so B to D to A to B? Then the second draw command is (10,0)--(20,0)--(8.2635,9.8481), which connects D(10,0) to C(20,0) to A(8.2635,9.8481) and back to D? So that's triangle ADC? Then there are some other lines. So overall, the diagram has triangle ABC with D on BC, and AD, BD, CD all equal. Therefore, D is midpoint of BC and circumradius. Therefore, ABC is right-angled at A. Therefore, angle BAC is 90 degrees. Therefore, answer is 90 degrees. But maybe the problem is in another configuration? Wait, but according to the Asymptote code, coordinates are given, so maybe the answer is 90 degrees. However, maybe I need to check with another method.

Alternatively, let me try to solve it without coordinates. Let me assume that AD = BD = CD. So D is the circumcenter. Therefore, angle BAC is equal to half the measure of the arc BC. Wait, no. In a circle, the central angle is twice the inscribed angle. Wait, if D is the circumcenter, then angle BDC is twice angle BAC. Wait, but since D is the circumcenter, BD and CD are radii. Wait, but if D is on BC, then BD = DC = AD. Therefore, triangle DBC is isoceles with BD = DC. Wait, but if D is on BC, then BD = DC implies that D is the midpoint of BC. Therefore, BC is the diameter of the circumcircle. Therefore, angle BAC is 90 degrees. Therefore, answer is 90 degrees. Therefore, regardless of other angles, angle BAC is 90 degrees. Therefore, answer is 90 degrees. Therefore, maybe the answer is 90 degrees. But why is angle BCA given as 40 degrees? Because in that case, angle at C is 40 degrees, angle at A is 90 degrees, angle at B is 50 degrees. Therefore, that's consistent. Therefore, answer is 90 degrees. Therefore, I think the answer is 90 degrees. But maybe I need to check again.

Wait, but maybe there's a mistake here. Suppose that D is not on BC? Wait, but according to the Asymptote code, D is on BC. But maybe in the problem, D is not on BC? Wait, the problem says "In the diagram, AD=BD=CD and angle BCA = 40 degrees." So the diagram is given, but since I don't have the diagram, but the Asymptote code shows D on BC. Therefore, assuming that, then answer is 90 degrees. However, maybe the problem is from an official source where the answer is different? Wait, but according to calculations, it's 90 degrees. Alternatively, maybe I misread the problem? Let me check again.

Wait, the problem says "AD=BD=CD". So D is equidistant from A, B, and C. Therefore, D is the circumcenter. Therefore, if D is the circumcenter, then ABC is a triangle with circumcenter D. If D is on BC, then BC is the diameter. Therefore, angle BAC is 90 degrees. Therefore, answer is 90 degrees. Therefore, I think that is correct. Therefore, maybe the answer is 90 degrees. However, maybe the problem is in three dimensions? No, it's a plane diagram. Therefore, I think the answer is 90 degrees. Therefore, I think the measure of angle BAC is 90 degrees. Therefore, boxed answer is \boxed{90}.

Wait, but let me check once more. Suppose angle BAC is 90 degrees, angle BCA is 40 degrees, then angle ABC is 50 degrees. Then, in triangle ABC, with BC as hypotenuse, then BD = DC = AD = R, the circumradius. In a right-angled triangle, the circumradius is half the hypotenuse. Therefore, BC = 2R. Therefore, BD = DC = R, which is half of BC. Therefore, D is the midpoint of BC. Therefore, yes, that's consistent. Therefore, all conditions are satisfied. Therefore, angle BAC is 90 degrees. Therefore, answer is 90 degrees. Therefore, I think that is correct.

**Final Answer**
\boxed{90}

To determine the measure of $\angle BAC$, we are given the following:

- $AD = BD = CD$
- $\angle BCA = 40^\circ$

---

### Step 1: Understanding the Geometry

The key condition is that $AD = BD = CD$. This implies that point $D$ is **equidistant** from points $A$, $B$, and $C$. In a triangle, the only point that is equidistant from all three vertices is the **circumcenter**. Therefore, point $D$ is the **circumcenter** of triangle $ABC$.

---

### Step 2: Properties of the Circumcenter

In any triangle, the circumcenter is the intersection of the perpendicular bisectors of the sides. It is also the center of the **circumcircle** of the triangle, and all three vertices lie on the circle centered at the circumcenter.

If the circumcenter lies on a side of the triangle (as in this case, since $D$ lies on segment $BC$), then that side must be the **diameter** of the circumcircle. This is a key geometric property: if a triangle is inscribed in a circle and one of its sides is the diameter, then the angle opposite that side is a **right angle**.

---

### Step 3: Applying the Right Angle Theorem

Since $D$ lies on segment $BC$ and is equidistant from $A$, $B$, and $C$, it must be the midpoint of $BC$, and thus $BC$ is the **diameter** of the circumcircle. Therefore, triangle $ABC$ is a **right triangle** with the right angle at point $A$.

This means:

$$
\angle BAC = 90^\circ
$$

---

### Step 4: Verifying with Given Angle

We are told that $\angle BCA = 40^\circ$. In a right triangle, the sum of the two non-right angles is $90^\circ$. Therefore, the remaining angle $\angle ABC = 50^\circ$, which is consistent with the triangle being right-angled at $A$.

---

### Final Conclusion

All conditions are satisfied with $\angle BAC = 90^\circ$, and the diagram confirms that $D$ is the midpoint of $BC$, making $BC$ the diameter of the circumcircle and $\angle BAC$ a right angle.

$$
\boxed{90}
$$

Solution:
Since $\angle BCA = 40^\circ$ and $\triangle ADC$ is isosceles with $AD=DC,$ we know $\angle DAC=\angle ACD=40^\circ.$

Since the sum of the angles in a triangle is $180^\circ,$ we have \begin{align*}
\angle ADC &= 180^\circ - \angle DAC - \angle ACD \\
&= 180^\circ - 40^\circ - 40^\circ \\
&= 100^\circ.
\end{align*}Since $\angle ADB$ and $\angle ADC$ are supplementary, we have  \begin{align*}
\angle ADB &= 180^\circ - \angle ADC \\
&= 180^\circ - 100^\circ \\
&= 80^\circ.
\end{align*}Since $\triangle ADB$ is isosceles with $AD=DB,$ we have $\angle BAD = \angle ABD.$ Thus, \begin{align*}
\angle BAD &= \frac{1}{2}(180^\circ - \angle ADB) \\
&= \frac{1}{2}(180^\circ - 80^\circ) \\
&= \frac{1}{2}(100^\circ) \\
&= 50^\circ.
\end{align*}Therefore,  \begin{align*}
\angle BAC &= \angle BAD + \angle DAC \\
&= 50^\circ+40^\circ \\
&= \boxed{90^\circ}.
\end{align*}


Explanation:
Okay, so I need to simplify this expression: 19x plus 1 minus 4x minus 81. Let me think about how to approach this. Alright, simplifying expressions usually involves combining like terms, right? So I should look for terms that have the same variable part or are just constants.

First, let me write down the expression again to make sure I have it right: 19x + 1 - 4x - 81. Yeah, that's what it says. So, the terms with x are 19x and -4x, and then the constant terms are +1 and -81.

Let me handle the x terms first. So 19x minus 4x. Hmm, 19 minus 4 is 15, so that should be 15x. Let me check that again. If I have 19x and subtract 4x, it's like 19x - 4x, which is (19 - 4)x = 15x. Yep, that seems right.

Now the constants: +1 and -81. So combining those, it's 1 - 81. Let me calculate that. 1 minus 81 is... hmm, that's like subtracting 81 from 1. So 1 - 81 equals -80. Wait, is that correct? Let me think. If I have 1 and then take away 81, that's like going into negative numbers. So 1 - 81 is indeed -80. Yeah, that's right.

So putting it all together, the simplified expression should be 15x - 80. Let me just verify that again. Original expression: 19x +1 -4x -81. Combine 19x and -4x: 15x. Combine +1 and -81: -80. So yes, 15x - 80.

Wait, is there anything else I need to do? Maybe check if there are any other like terms or if I can factor anything out? Let me see. 15x and -80 don't have any common factors except maybe 5? Wait, 15 and 80 have a common factor of 5. But since they are separate terms, unless there's a way to factor something out, but the question just says to simplify the expression. So combining like terms is the main step here.

So I think that's it. The simplified form is 15x - 80. Let me just write that again to make sure I didn't make any arithmetic errors.

Original terms: 19x - 4x = 15x. Constants: 1 - 81 = -80. Yep, that's correct. So the answer is 15x - 80.

I don't think I made any mistakes here. Let me try plugging in a value for x to check. Let's say x = 0. Original expression: 19*0 +1 -4*0 -81 = 0 +1 -0 -81 = -80. Simplified expression: 15*0 -80 = 0 -80 = -80. Same result.

Another test: x = 1. Original: 19*1 +1 -4*1 -81 = 19 +1 -4 -81 = (19 +1) = 20; (20 -4) = 16; (16 -81) = -65. Simplified: 15*1 -80 = 15 -80 = -65. Same result.

Another test: x = 2. Original: 19*2 +1 -4*2 -81 = 38 +1 -8 -81 = 39 -8 = 31; 31 -81 = -50. Simplified: 15*2 -80 = 30 -80 = -50. Same again.

Okay, so that seems to check out. So I think my simplification is correct. Therefore, the simplified expression is 15x - 80.

**Final Answer**
The simplified expression is \boxed{15x - 80}.

To simplify the expression:

$$
19x + 1 - 4x - 81
$$

we follow a systematic approach by combining like terms.

---

### Step 1: Combine the x-terms

The terms involving $ x $ are:

$$
19x \quad \text{and} \quad -4x
$$

Adding these together:

$$
19x - 4x = (19 - 4)x = 15x
$$

---

### Step 2: Combine the constant terms

The constant terms are:

$$
1 \quad \text{and} \quad -81
$$

Adding these together:

$$
1 - 81 = -80
$$

---

### Step 3: Write the simplified expression

Now, combining the results from both steps:

$$
15x - 80
$$

---

### Final Answer

$$
\boxed{15x - 80}
$$

[Assistant]
Explanation:
Okay, so I need to simplify this expression: 19x plus 1 minus 4x minus 81. Let me think about how to approach this. Alright, simplifying expressions usually involves combining like terms, right? So I should look for terms that have the same variable part or are just constants.

First, let me write down the expression again to make sure I have it right: 19x + 1 - 4x - 81. Yeah, that's what it says. So, the terms with x are 19x and -4x, and then the constant terms are +1 and -81.

Let me handle the x terms first. So 19x minus 4x. Hmm, 19 minus 4 is 15, so that should be 15x. Let me check that again. If I have 19x and subtract 4x, it's like 19x - 4x, which is (19 - 4)x = 15x. Yep, that seems right.

Now the constants: +1 and -81. So combining those, it's 1 - 81. Let me calculate that. 1 minus 81 is... hmm, that's like subtracting 81 from 1. So 1 - 81 equals -80. Wait, is that correct? Let me think. If I have 1 and then take away 81, that's like going into negative numbers. So 1 - 81 is indeed -80. Yeah, that's right.

So putting it all together, the simplified expression should be 15x - 80. Let me just verify that again. Original expression: 19x +1 -4x -81. Combine 19x and -4x: 15x. Combine +1 and -81: -80. So yes, 15x - 80.

Wait, is there anything else I need to do? Maybe check if there are any other like terms or if I can factor anything out? Let me see. 15x and -80 don't have any common factors except maybe 5? Wait, 15 and 80 have a common factor of 5. But since they are separate terms, unless there's a way to factor something out, but the question just says to simplify the expression. So combining like terms is the main step here.

So I think that's it. The simplified form is 15x - 80. Let me just write that again to make sure I didn't make any arithmetic errors.

Original terms: 19x - 4x = 15x. Constants: 1 - 81 = -80. Yep, that's correct. So the answer is 15x - 80.

I don't think I made any mistakes here. Let me try plugging in a value for x to check. Let's say x = 0. Original expression: 19*0 +1 -4*0 -81 = 0 +1 -0 -81 = -80. Simplified expression: 15*0 -80 = 0 -80 = -80. Same result.

Another test: x = 1. Original: 19*1 +1 -4*1 -81 = 19 +1 -4 -81 = (19 +1) = 20; (20 -4) = 16; (16 -81) = -65. Simplified: 15*1 -80 = 15 -80 = -65. Same result.

Another test: x = 2. Original: 19*2 +1 -4*2 -81 = 38 +1 -8 -81 = 39 -8 = 31; 31 -81 = -50. Simplified: 15*2 -80 = 30 -80 = -50. Same again.

Okay, so that seems to check out. So I think my simplification is correct. Therefore, the simplified expression is 15x - 80.

**Final Answer**
The simplified expression is \boxed{15x - 80}.

To simplify the expression:

$$
19x + 1 - 4x - 81
$$

we follow a systematic approach by combining like terms.

---

### Step 1: Combine the x-terms

The terms involving $ x $ are:

$$
19x \quad \text{and} \quad -4x
$$

Adding these together:

$$
19x - 4x = (19 - 4)x = 15x
$$

---

### Step 2: Combine the constant terms

The constant terms are:

$$
1 \quad \text{and} \quad -81
$$

Adding these together:

$$
1 - 81 = -80
$$

---

### Step 3: Write the simplified expression

Now, combining the results from both steps:

$$
15x - 80
$$

---

### Final Answer

$$
\boxed{15x - 80}
$$

Solution: Rearranging and grouping, we obtain $(19x - 4x) + (1 - 81) = \boxed{15x - 80}$.


Explanation:
Okay, so I need to solve this equation: 3x plus 2 times (1 plus x) equals 17. And then find the value of 6x plus 5. Let me think about how to approach this step by step.

First, let me write down the equation again to make sure I have it right: 3x + 2(1 + x) = 17. Alright, so there's a distributive property here with the 2 multiplied by (1 + x). Maybe I should start by distributing that 2 into the parentheses. Let me do that.

So, 2 times 1 is 2, and 2 times x is 2x. So, the equation becomes 3x + 2 + 2x = 17. Wait, let me check that again. Yes, distributing the 2 gives me 2*1 and 2*x, so that's 2 + 2x. So combining that with the original 3x, the equation is now 3x + 2x + 2 = 17.

Now, combining like terms. The 3x and 2x can be added together. 3x + 2x is 5x. So now the equation is 5x + 2 = 17. Okay, that seems simpler.

Now, I need to solve for x. To isolate x, I should get rid of the 2 that's being added to 5x. So, subtract 2 from both sides of the equation. Let me do that: 5x + 2 - 2 = 17 - 2. Simplifying both sides, that gives me 5x = 15.

Now, to solve for x, I need to divide both sides by 5. So, 5x divided by 5 is x, and 15 divided by 5 is 3. Therefore, x equals 3. Let me check if that makes sense.

Let me substitute x = 3 back into the original equation to verify. Original equation: 3x + 2(1 + x) = 17. Plugging in x = 3: 3*3 + 2*(1 + 3). Calculating each part: 3*3 is 9, and 1 + 3 is 4, so 2*4 is 8. Adding those together: 9 + 8 = 17. Which matches the right side of the equation. Perfect, so x = 3 is correct.

Now, the question asks for the value of 6x + 5. Since I know x is 3, I can substitute that in. Let me compute 6*3 + 5. 6 times 3 is 18, and 18 plus 5 is 23. So, 6x + 5 equals 23.

Wait, let me make sure I didn't make any mistakes here. Let me go through the steps again.

Original equation: 3x + 2(1 + x) = 17. Distribute the 2: 3x + 2 + 2x = 17. Combine like terms: 5x + 2 = 17. Subtract 2: 5x = 15. Divide by 5: x = 3. Then 6x + 5 is 6*3 + 5 = 18 + 5 = 23. Seems straightforward.

Alternatively, maybe there's another way to approach the problem? Let me think. Suppose instead of solving for x first, could I find 6x + 5 directly from the original equation? Let me see.

Original equation: 3x + 2(1 + x) = 17. Let me expand that again: 3x + 2 + 2x = 17, which is 5x + 2 = 17. So, 5x = 15, so x = 3. So, same result.

Alternatively, if I wanted to find 6x + 5, maybe I can express that in terms of the original equation. Let me see. Let's see, 6x + 5. If I have 5x = 15 from earlier, then 6x would be 5x + x, which is 15 + x. Therefore, 6x + 5 would be 15 + x + 5 = 20 + x. Since x is 3, that would be 20 + 3 = 23. Same answer.

Alternatively, maybe manipulate the original equation to get 6x + 5. Let me try that.

Original equation: 3x + 2(1 + x) = 17. Let me expand that again: 3x + 2 + 2x = 17. Combine terms: 5x + 2 = 17. Let me rearrange this equation. If I subtract 5x from both sides, I get 2 = 17 - 5x. But not sure if that helps.

Alternatively, maybe express 6x + 5 in terms of 5x. Since 5x is 15, then 6x is 5x + x, so 15 + x. Then 6x + 5 is 15 + x + 5 = 20 + x. Which again, since x is 3, gives 23.

Alternatively, maybe think of 6x + 5 as 2*(3x) + 5. Wait, original equation has 3x. Let me see. If I take the original equation: 3x + 2(1 + x) = 17. Let me solve for 3x.

From 3x + 2(1 + x) = 17, we can write 3x = 17 - 2(1 + x). Let me compute that: 17 - 2 - 2x = 15 - 2x. Therefore, 3x = 15 - 2x. Then, adding 2x to both sides: 5x = 15, so x = 3. Again, same result.

But if I want to find 6x + 5, maybe express it in terms of 3x. Since 6x is 2*(3x). So, 6x + 5 = 2*(3x) + 5. From above, 3x = 15 - 2x. Therefore, substituting that in: 2*(15 - 2x) + 5 = 30 - 4x + 5 = 35 - 4x. Hmm, but then I need to find 35 - 4x. Since x is 3, that would be 35 - 12 = 23. Same answer.

So, regardless of the method, I end up with 23. Therefore, I think that's the correct answer.

But just to make sure, let me check once again. If x is 3, then 6x + 5 is 6*3 + 5. 6*3 is 18, 18 + 5 is 23. Yep.

Alternatively, maybe there's a mistake in the initial steps? Let me check the original equation again.

Original equation: 3x + 2(1 + x) = 17. Let me compute left-hand side with x = 3. 3*3 is 9. 1 + 3 is 4. 2*4 is 8. 9 + 8 is 17. Which equals the right-hand side. Perfect.

So, all steps check out. Therefore, the value of 6x + 5 is 23.

I think that's solid. I don't see any errors in my reasoning. So, I can confidently say the answer is 23.

**Final Answer**
The value of $6x + 5$ is \boxed{23}.

To solve the equation:

$$
3x + 2(1 + x) = 17
$$

we begin by simplifying the left-hand side using the distributive property:

$$
3x + 2(1 + x) = 3x + 2 + 2x
$$

Now, combine like terms:

$$
3x + 2x + 2 = 5x + 2
$$

So the equation becomes:

$$
5x + 2 = 17
$$

---

### Step 1: Solve for $ x $

Subtract 2 from both sides:

$$
5x = 15
$$

Now divide both sides by 5:

$$
x = 3
$$

---

### Step 2: Find the value of $ 6x + 5 $

Substitute $ x = 3 $ into the expression:

$$
6x + 5 = 6(3) + 5 = 18 + 5 = 23
$$

---

### Final Answer

$$
\boxed{23}
$$

[Assistant]
Explanation:
Okay, so I need to solve this equation: 3x plus 2 times (1 plus x) equals 17. And then find the value of 6x plus 5. Let me think about how to approach this step by step.

First, let me write down the equation again to make sure I have it right: 3x + 2(1 + x) = 17. Alright, so there's a distributive property here with the 2 multiplied by (1 + x). Maybe I should start by distributing that 2 into the parentheses. Let me do that.

So, 2 times 1 is 2, and 2 times x is 2x. So, the equation becomes 3x + 2 + 2x = 17. Wait, let me check that again. Yes, distributing the 2 gives me 2*1 and 2*x, so that's 2 + 2x. So combining that with the original 3x, the equation is now 3x + 2x + 2 = 17.

Now, combining like terms. The 3x and 2x can be added together. 3x + 2x is 5x. So now the equation is 5x + 2 = 17. Okay, that seems simpler.

Now, I need to solve for x. To isolate x, I should get rid of the 2 that's being added to 5x. So, subtract 2 from both sides of the equation. Let me do that: 5x + 2 - 2 = 17 - 2. Simplifying both sides, that gives me 5x = 15.

Now, to solve for x, I need to divide both sides by 5. So, 5x divided by 5 is x, and 15 divided by 5 is 3. Therefore, x equals 3. Let me check if that makes sense.

Let me substitute x = 3 back into the original equation to verify. Original equation: 3x + 2(1 + x) = 17. Plugging in x = 3: 3*3 + 2*(1 + 3). Calculating each part: 3*3 is 9, and 1 + 3 is 4, so 2*4 is 8. Adding those together: 9 + 8 = 17. Which matches the right side of the equation. Perfect, so x = 3 is correct.

Now, the question asks for the value of 6x + 5. Since I know x is 3, I can substitute that in. Let me compute 6*3 + 5. 6 times 3 is 18, and 18 plus 5 is 23. So, 6x + 5 equals 23.

Wait, let me make sure I didn't make any mistakes here. Let me go through the steps again.

Original equation: 3x + 2(1 + x) = 17. Distribute the 2: 3x + 2 + 2x = 17. Combine like terms: 5x + 2 = 17. Subtract 2: 5x = 15. Divide by 5: x = 3. Then 6x + 5 is 6*3 + 5 = 18 + 5 = 23. Seems straightforward.

Alternatively, maybe there's another way to approach the problem? Let me think. Suppose instead of solving for x first, could I find 6x + 5 directly from the original equation? Let me see.

Original equation: 3x + 2(1 + x) = 17. Let me expand that again: 3x + 2 + 2x = 17, which is 5x + 2 = 17. So, 5x = 15, so x = 3. So, same result.

Alternatively, if I wanted to find 6x + 5, maybe I can express that in terms of the original equation. Let me see. Let's see, 6x + 5. If I have 5x = 15 from earlier, then 6x would be 5x + x, which is 15 + x. Therefore, 6x + 5 would be 15 + x + 5 = 20 + x. Since x is 3, that would be 20 + 3 = 23. Same answer.

Alternatively, maybe manipulate the original equation to get 6x + 5. Let me try that.

Original equation: 3x + 2(1 + x) = 17. Let me expand that again: 3x + 2 + 2x = 17. Combine terms: 5x + 2 = 17. Let me rearrange this equation. If I subtract 5x from both sides, I get 2 = 17 - 5x. But not sure if that helps.

Alternatively, maybe express 6x + 5 in terms of 5x. Since 5x is 15, then 6x is 5x + x, so 15 + x. Then 6x + 5 is 15 + x + 5 = 20 + x. Which again, since x is 3, gives 23.

Alternatively, maybe think of 6x + 5 as 2*(3x) + 5. Wait, original equation has 3x. Let me see. If I take the original equation: 3x + 2(1 + x) = 17. Let me solve for 3x.

From 3x + 2(1 + x) = 17, we can write 3x = 17 - 2(1 + x). Let me compute that: 17 - 2 - 2x = 15 - 2x. Therefore, 3x = 15 - 2x. Then, adding 2x to both sides: 5x = 15, so x = 3. Again, same result.

But if I want to find 6x + 5, maybe express it in terms of 3x. Since 6x is 2*(3x). So, 6x + 5 = 2*(3x) + 5. From above, 3x = 15 - 2x. Therefore, substituting that in: 2*(15 - 2x) + 5 = 30 - 4x + 5 = 35 - 4x. Hmm, but then I need to find 35 - 4x. Since x is 3, that would be 35 - 12 = 23. Same answer.

So, regardless of the method, I end up with 23. Therefore, I think that's the correct answer.

But just to make sure, let me check once again. If x is 3, then 6x + 5 is 6*3 + 5. 6*3 is 18, 18 + 5 is 23. Yep.

Alternatively, maybe there's a mistake in the initial steps? Let me check the original equation again.

Original equation: 3x + 2(1 + x) = 17. Let me compute left-hand side with x = 3. 3*3 is 9. 1 + 3 is 4. 2*4 is 8. 9 + 8 is 17. Which equals the right-hand side. Perfect.

So, all steps check out. Therefore, the value of 6x + 5 is 23.

I think that's solid. I don't see any errors in my reasoning. So, I can confidently say the answer is 23.

**Final Answer**
The value of $6x + 5$ is \boxed{23}.

To solve the equation:

$$
3x + 2(1 + x) = 17
$$

we begin by simplifying the left-hand side using the distributive property:

$$
3x + 2(1 + x) = 3x + 2 + 2x
$$

Now, combine like terms:

$$
3x + 2x + 2 = 5x + 2
$$

So the equation becomes:

$$
5x + 2 = 17
$$

---

### Step 1: Solve for $ x $

Subtract 2 from both sides:

$$
5x = 15
$$

Now divide both sides by 5:

$$
x = 3
$$

---

### Step 2: Find the value of $ 6x + 5 $

Substitute $ x = 3 $ into the expression:

$$
6x + 5 = 6(3) + 5 = 18 + 5 = 23
$$

---

### Final Answer

$$
\boxed{23}
$$

Solution: Expanding and collecting terms on the left hand side of the first equation gives $5x+2=17$. Subtracting 2 from each side gives $5x=15$, then dividing each side by 5 gives $x=3$. Now that we know what $x$ is, we can substitute it into $6x+5$ and get $6(3)+5=18+5=\boxed{23}$.


Explanation:
Okay, so I need to add 313.9 and 12.6 together. Let me think about how to do this. Hmm, adding decimals... I remember that you have to line up the decimal points, right? So maybe I should write them down vertically to make sure I add each place value correctly. Let me try that.

First, let me write 313.9. Since 12.6 has one decimal place, I should make sure both numbers have the same number of decimal places. Wait, 313.9 is the same as 313.90, right? So maybe I can write it like that to align the decimals properly. Let me try:

  313.90
+  12.60
________

Wait, but actually, 12.6 is the same as 12.60, so adding them together. Let me check if that's correct. Yeah, adding zeros to the end of a decimal doesn't change its value. So that should be okay.

Now, let me add them column by column, starting from the rightmost digit. Let me write it out:

Starting with the hundredths place: 0 + 0 = 0. Then the tenths place: 9 + 6 = 15. Oh, wait, that's 15 tenths. So I need to carry over 1 to the units place. Let me write that down.

So tenths place: 9 + 6 = 15. So I put down 5 and carry over 1. Then moving to the units place: 3 (from 313.90) + 2 (from 12.60) + the carried over 1. Let me check. Wait, the units place of 313.90 is 3, and the units place of 12.60 is 2. So 3 + 2 = 5, plus the carried over 1 makes 6. So units place is 6.

Next, the tens place: 1 (from 313.90) + 1 (from 12.60) = 2. Wait, hold on. Let me make sure I'm aligning the digits correctly. Let me write the numbers again:

  313.90
+  12.60
________

Breaking it down:

- The digits to the right of the decimal: 90 and 60. So adding 90 + 60 = 150. So that's 1.50? Wait, no. Wait, actually, when adding decimals, it's better to do each column step by step.

Let me try again. Let me write each digit:

Starting from the rightmost digit (hundredths place):

First number: 313.90 has digits: 3 (hundreds), 1 (tens), 3 (units), ., 9 (tenths), 0 (hundredths).

Second number: 12.60 has digits: 0 (hundreds), 1 (tens), 2 (units), ., 6 (tenths), 0 (hundredths).

So adding them:

Hundredths place: 0 + 0 = 0.

Tenths place: 9 + 6 = 15. So write down 5, carry over 1.

Units place: 3 + 2 = 5, plus the carried over 1 makes 6.

Tens place: 1 + 1 = 2.

Hundreds place: 3 + 0 = 3.

So putting it all together: 3 (hundreds), 2 (tens), 6 (units), ., 5 (tenths), 0 (hundredths). So that would be 326.50.

Wait, but 313.9 + 12.6. Let me check with another method. Maybe convert them to fractions?

313.9 is the same as 313 + 0.9. 0.9 is 9/10. Similarly, 12.6 is 12 + 6/10. So adding them together:

313 + 12 = 325, and 9/10 + 6/10 = 15/10 = 1 5/10. So adding that to 325 gives 325 + 1 5/10 = 326 5/10, which is 326.5. So that's 326.5. But earlier, when I added the decimals, I got 326.50. Which is the same as 326.5. So both methods give the same result. So that seems correct.

Alternatively, maybe I can just add them directly without aligning decimals. Let me try that. 313.9 + 12.6. Let me think of 313.9 + 12.6. Adding the whole numbers first: 313 + 12 = 325. Then adding the decimals: 0.9 + 0.6 = 1.5. Then adding those together: 325 + 1.5 = 326.5. Yep, same result. So that's 326.5.

So the answer should be 326.5. Let me check once more with another approach. Let me use a calculator in my mind. 313.9 + 12.6. Let me add 313.9 + 10 = 323.9, then add 2.6 more. 323.9 + 2.6. 323.9 + 2 = 325.9, then + 0.6 = 326.5. Yep, same answer. So that's consistent.

Therefore, the result of adding 313.9 and 12.6 is 326.5. So expressed as a decimal, it's 326.5. I think that's the correct answer. Let me just make sure there are no mistakes in my steps. All methods lead to 326.5, so I think that's right.

**Final Answer**
The result of adding 313.9 and 12.6 is \boxed{326.5}.

To solve the problem of adding 313.9 and 12.6, we follow a systematic approach to ensure accuracy in decimal arithmetic.

---

### Step 1: Align the Decimal Points

To add decimals, it's important to align the numbers by their decimal points. This ensures that we are adding digits of the same place value (tenths, units, tens, etc.).

We can write the numbers as:

```
  313.90
+  12.60
```

This alignment helps us add each column correctly.

---

### Step 2: Add Column by Column

We now add the digits from right to left, starting with the hundredths place:

- **Hundredths place**: 0 + 0 = 0
- **Tenths place**: 9 + 6 = 15 → write down 5, carry over 1
- **Units place**: 3 + 2 + 1 (carry) = 6
- **Tens place**: 1 + 1 = 2
- **Hundreds place**: 3 + 0 = 3

Putting it all together, we get:

```
  313.90
+  12.60
--------
  326.50
```

---

### Step 3: Simplify the Result

The result is 326.50, which is equivalent to 326.5 when expressed as a decimal. The trailing zero after the decimal point is not necessary and can be omitted.

---

### Final Answer

$$
\boxed{326.5}
$$

[Assistant]
Explanation:
Okay, so I need to add 313.9 and 12.6 together. Let me think about how to do this. Hmm, adding decimals... I remember that you have to line up the decimal points, right? So maybe I should write them down vertically to make sure I add each place value correctly. Let me try that.

First, let me write 313.9. Since 12.6 has one decimal place, I should make sure both numbers have the same number of decimal places. Wait, 313.9 is the same as 313.90, right? So maybe I can write it like that to align the decimals properly. Let me try:

  313.90
+  12.60
________

Wait, but actually, 12.6 is the same as 12.60, so adding them together. Let me check if that's correct. Yeah, adding zeros to the end of a decimal doesn't change its value. So that should be okay.

Now, let me add them column by column, starting from the rightmost digit. Let me write it out:

Starting with the hundredths place: 0 + 0 = 0. Then the tenths place: 9 + 6 = 15. Oh, wait, that's 15 tenths. So I need to carry over 1 to the units place. Let me write that down.

So tenths place: 9 + 6 = 15. So I put down 5 and carry over 1. Then moving to the units place: 3 (from 313.90) + 2 (from 12.60) + the carried over 1. Let me check. Wait, the units place of 313.90 is 3, and the units place of 12.60 is 2. So 3 + 2 = 5, plus the carried over 1 makes 6. So units place is 6.

Next, the tens place: 1 (from 313.90) + 1 (from 12.60) = 2. Wait, hold on. Let me make sure I'm aligning the digits correctly. Let me write the numbers again:

  313.90
+  12.60
________

Breaking it down:

- The digits to the right of the decimal: 90 and 60. So adding 90 + 60 = 150. So that's 1.50? Wait, no. Wait, actually, when adding decimals, it's better to do each column step by step.

Let me try again. Let me write each digit:

Starting from the rightmost digit (hundredths place):

First number: 313.90 has digits: 3 (hundreds), 1 (tens), 3 (units), ., 9 (tenths), 0 (hundredths).

Second number: 12.60 has digits: 0 (hundreds), 1 (tens), 2 (units), ., 6 (tenths), 0 (hundredths).

So adding them:

Hundredths place: 0 + 0 = 0.

Tenths place: 9 + 6 = 15. So write down 5, carry over 1.

Units place: 3 + 2 = 5, plus the carried over 1 makes 6.

Tens place: 1 + 1 = 2.

Hundreds place: 3 + 0 = 3.

So putting it all together: 3 (hundreds), 2 (tens), 6 (units), ., 5 (tenths), 0 (hundredths). So that would be 326.50.

Wait, but 313.9 + 12.6. Let me check with another method. Maybe convert them to fractions?

313.9 is the same as 313 + 0.9. 0.9 is 9/10. Similarly, 12.6 is 12 + 6/10. So adding them together:

313 + 12 = 325, and 9/10 + 6/10 = 15/10 = 1 5/10. So adding that to 325 gives 325 + 1 5/10 = 326 5/10, which is 326.5. So that's 326.5. But earlier, when I added the decimals, I got 326.50. Which is the same as 326.5. So both methods give the same result. So that seems correct.

Alternatively, maybe I can just add them directly without aligning decimals. Let me try that. 313.9 + 12.6. Let me think of 313.9 + 12.6. Adding the whole numbers first: 313 + 12 = 325. Then adding the decimals: 0.9 + 0.6 = 1.5. Then adding those together: 325 + 1.5 = 326.5. Yep, same result. So that's 326.5.

So the answer should be 326.5. Let me check once more with another approach. Let me use a calculator in my mind. 313.9 + 12.6. Let me add 313.9 + 10 = 323.9, then add 2.6 more. 323.9 + 2.6. 323.9 + 2 = 325.9, then + 0.6 = 326.5. Yep, same answer. So that's consistent.

Therefore, the result of adding 313.9 and 12.6 is 326.5. So expressed as a decimal, it's 326.5. I think that's the correct answer. Let me just make sure there are no mistakes in my steps. All methods lead to 326.5, so I think that's right.

**Final Answer**
The result of adding 313.9 and 12.6 is \boxed{326.5}.

To solve the problem of adding 313.9 and 12.6, we follow a systematic approach to ensure accuracy in decimal arithmetic.

---

### Step 1: Align the Decimal Points

To add decimals, it's important to align the numbers by their decimal points. This ensures that we are adding digits of the same place value (tenths, units, tens, etc.).

We can write the numbers as:

```
  313.90
+  12.60
```

This alignment helps us add each column correctly.

---

### Step 2: Add Column by Column

We now add the digits from right to left, starting with the hundredths place:

- **Hundredths place**: 0 + 0 = 0
- **Tenths place**: 9 + 6 = 15 → write down 5, carry over 1
- **Units place**: 3 + 2 + 1 (carry) = 6
- **Tens place**: 1 + 1 = 2
- **Hundreds place**: 3 + 0 = 3

Putting it all together, we get:

```
  313.90
+  12.60
--------
  326.50
```

---

### Step 3: Simplify the Result

The result is 326.50, which is equivalent to 326.5 when expressed as a decimal. The trailing zero after the decimal point is not necessary and can be omitted.

---

### Final Answer

$$
\boxed{326.5}
$$

Solution:
We have \[
\begin{array}{@{}c@{}c@{}c@{}c@{}c@{}c}
& & & 1 & \\
& 3 & 1 & 3. & 9 \\
+ & & 1 & 2. & 6
\\ \cline{1-5}
& 3 & 2 & 6. & 5 \\
\end{array}
\]Since $9+6=15$ in the right-most column is greater than 9, we record the 5 and "carry" the 10 to the next column as a 1 (shown over the second 3 in the addend 313.9). The answer is $\boxed{326.5}$.


Explanation:
Okay, so I need to figure out the units digit N of the number 21420N such that the entire number is divisible by 6. And N has to be a nonzero digit. Let me start by recalling the rules for divisibility by 6.

Divisibility by 6 means the number has to be divisible by both 2 and 3, right? Because 6 is 2 times 3, and 2 and 3 are prime numbers. So, the number must satisfy both divisibility rules.

First, let me remember the divisibility rules. For a number to be divisible by 2, its last digit (the units digit) has to be even. For divisibility by 3, the sum of all its digits must be a multiple of 3.

So, since the number is 21420N, the units digit is N. Therefore, for divisibility by 2, N must be even. That narrows down the possible values of N to 2, 4, 6, or 8 because N is a nonzero digit.

Now, for divisibility by 3, I need to calculate the sum of all the digits of the number 21420N and check if that sum is divisible by 3. Let me write down the digits: 2, 1, 4, 2, 0, and then N.

Let me add them up. Let's compute the sum step by step.

Starting with 2 + 1 = 3. Then 3 + 4 = 7. Then 7 + 2 = 9. Then 9 + 0 = 9. Then adding N, the total sum is 9 + N.

So, the sum of the digits is 9 + N. For the number to be divisible by 3, 9 + N must be a multiple of 3.

Now, since 9 is already a multiple of 3 (because 9 divided by 3 is 3), adding N to it will result in a multiple of 3 if N is also a multiple of 3. Wait, let me think again.

If 9 is divisible by 3, then adding N to it will make the total sum divisible by 3 if N is divisible by 3. Because if you have a number divisible by 3 and add another number divisible by 3, the result is still divisible by 3. Conversely, if N is not divisible by 3, then the sum won't be.

But let me check with an example. Suppose N is 0: 9 + 0 = 9, which is divisible by 3. But N can't be 0 because the problem says nonzero. If N is 1: 9 + 1 = 10, which is not divisible by 3. N=2: 9 + 2 = 11, not divisible by 3. N=3: 9 + 3 = 12, which is divisible by 3. N=4: 13, not. N=5: 14, nope. N=6: 15, yes. N=7: 16, nope. N=8: 17, nope. N=9: 18, yes.

But remember, N has to be even because of divisibility by 2. So from the earlier list of possible N (2,4,6,8), which of these make 9 + N divisible by 3? Let's check:

For N=2: 9 + 2 = 11. 11 divided by 3 is 3 with a remainder of 2. Not divisible.

N=4: 9 + 4 = 13. 13 divided by 3 is 4 with remainder 1. Not divisible.

N=6: 9 + 6 = 15. 15 divided by 3 is 5. Perfect, divisible by 3.

N=8: 9 + 8 = 17. 17 divided by 3 is 5 with remainder 2. Not divisible.

So only N=6 satisfies both conditions: it's even (so divisible by 2) and 9 + 6 = 15 is divisible by 3. Therefore, N=6 is the answer.

Wait, but let me double-check. Let me write out the number 214206 and check if it's divisible by 6.

First, check divisibility by 2: the last digit is 6, which is even. Good.

Divisibility by 3: sum of digits is 2 + 1 + 4 + 2 + 0 + 6. Let me compute that again. 2 + 1 is 3, plus 4 is 7, plus 2 is 9, plus 0 is 9, plus 6 is 15. 15 is divisible by 3. So yes, 214206 is divisible by 3. Therefore, it's divisible by 6.

Is there another possible N? Let me check again. The possible Ns are 2,4,6,8. For N=6, it works. What about N=3? Wait, N=3 is not even, so it's excluded. Similarly, N=9 is not even. So only N=6.

But let me check if there's a mistake in my calculation. Maybe I miscalculated the sum of digits? Let me check again. The original number is 21420N. So digits are 2, 1, 4, 2, 0, N. So adding them up: 2 + 1 is 3, plus 4 is 7, plus 2 is 9, plus 0 is 9, plus N. So yes, 9 + N.

So 9 + N must be divisible by 3. Since 9 is already divisible by 3, N must be divisible by 3. But N is also even. So N must be a digit that is both even and divisible by 3. The digits that are even are 0,2,4,6,8. Among these, which are divisible by 3? Well, 0 is divisible by 3, but N is nonzero. So 6 is the only one. Because 6 divided by 3 is 2. 0 is excluded. So yes, N=6.

Therefore, the answer is 6.

But let me check if there's another way to approach this. Maybe by considering the entire number. Let me think.

The number is 21420N. Let me think of it as 214200 + N. Wait, no. Wait, 21420N is a 6-digit number? Wait, hold on. Wait, the original number is written as 21420N. Wait, is that 21420 followed by N? So if N is a single digit, then the number is 21420N, which would be a 6-digit number. For example, if N is 6, it's 214206.

But maybe the original problem is written as 21420N, where N is the units digit. So the number is 21420N, meaning that the original number is 21420 with N as the last digit. So yes, that's a 6-digit number.

Alternatively, maybe the problem is written as 21420N, where N is a digit, so the number is 21420N. So, for example, if N is 6, then the number is 214206.

But maybe I need to check if there's a different interpretation. But I think that's the correct way.

Alternatively, maybe the problem is written as 21420N, where N is a digit, so the entire number is 21420N. So, for example, if N is 6, then the number is 214206.

So, in that case, the number is 21420N.

But maybe I can check divisibility by 6 by using the actual number. Let me try with N=6.

214206 divided by 6. Let me compute that.

First, divide by 2: 214206 / 2 = 107103. Then divide by 3: 107103 / 3. Let me compute that.

Divide 107103 by 3: 3 goes into 10 three times (3*3=9), remainder 1. Bring down 7: 17. 3 into 17 is 5 times (15), remainder 2. Bring down 1: 21. 3 into 21 is 7. Bring down 0: 0. 3 into 0 is 0. Bring down 3: 3 into 3 is 1. So the result is 35701. So 3*35701=107103. Therefore, 214206 divided by 6 is 35701, which is an integer. Therefore, yes, it is divisible by 6.

Alternatively, if I tried N=2: 214202. Let me check divisibility by 6. Divisible by 2? Yes, ends with 2. Divisible by 3? Sum of digits: 2+1+4+2+0+2= 11. 11 is not divisible by 3. Therefore, not divisible by 3.

N=4: sum is 2+1+4+2+0+4= 13, not divisible by 3.

N=6: sum is 15, which is divisible by 3.

N=8: sum is 17, not divisible by 3.

So only N=6 works.

Therefore, the answer is 6.

But let me check once more if there's a mistake in my logic. For example, maybe I miscalculated the sum of digits? Let me check again.

Original number: 2, 1, 4, 2, 0, N.

Adding them up: 2 + 1 = 3; 3 + 4 = 7; 7 + 2 = 9; 9 + 0 = 9; 9 + N.

Yes, that's correct. So 9 + N.

So 9 + N must be divisible by 3. Since 9 is already divisible by 3, N must be divisible by 3.

But N is also even. So N must be even and divisible by 3. The digits that are even and divisible by 3 are 0 and 6. Since N is nonzero, only 6.

Therefore, N=6.

Therefore, the answer is 6.

I think that's solid. I don't see any mistakes in my reasoning. So I think the correct answer is 6.

**Final Answer**
The nonzero value of $ N $ that makes the number divisible by 6 is \boxed{6}.

To determine the nonzero value of $ N $ that makes the number $ 21420N $ divisible by 6, we need to apply the rules of divisibility by 6. A number is divisible by 6 if and only if it is divisible by both 2 and 3.

---

### Step 1: Divisibility by 2

A number is divisible by 2 if its **last digit is even**. In this case, the last digit is $ N $. Therefore, $ N $ must be an even digit.

The possible even digits are:
$$
N \in \{2, 4, 6, 8\}
$$

---

### Step 2: Divisibility by 3

A number is divisible by 3 if the **sum of its digits is divisible by 3**. The digits of the number $ 21420N $ are:
$$
2, 1, 4, 2, 0, N
$$

Adding these digits:
$$
2 + 1 + 4 + 2 + 0 + N = 9 + N
$$

So, the sum of the digits is $ 9 + N $. For the number to be divisible by 3, $ 9 + N $ must be divisible by 3.

Since 9 is already divisible by 3, $ N $ must also be divisible by 3 for the sum to remain divisible by 3.

---

### Step 3: Combine Both Conditions

From Step 1, $ N $ must be even:
$$
N \in \{2, 4, 6, 8\}
$$

From Step 2, $ N $ must be divisible by 3:
$$
N \in \{3, 6, 9\}
$$

The only digit that satisfies **both** conditions is:
$$
N = 6
$$

---

### Step 4: Final Verification

Let’s verify that $ N = 6 $ makes the number $ 214206 $ divisible by 6:

- **Divisibility by 2**: The last digit is 6, which is even ✅
- **Divisibility by 3**: Sum of digits = $ 2 + 1 + 4 + 2 + 0 + 6 = 15 $, and 15 is divisible by 3 ✅

Thus, the number is divisible by both 2 and 3, and therefore by 6.

---

### Final Answer

$$
\boxed{6}
$$

[Assistant]
Explanation:
Okay, so I need to figure out the units digit N of the number 21420N such that the entire number is divisible by 6. And N has to be a nonzero digit. Let me start by recalling the rules for divisibility by 6.

Divisibility by 6 means the number has to be divisible by both 2 and 3, right? Because 6 is 2 times 3, and 2 and 3 are prime numbers. So, the number must satisfy both divisibility rules.

First, let me remember the divisibility rules. For a number to be divisible by 2, its last digit (the units digit) has to be even. For divisibility by 3, the sum of all its digits must be a multiple of 3.

So, since the number is 21420N, the units digit is N. Therefore, for divisibility by 2, N must be even. That narrows down the possible values of N to 2, 4, 6, or 8 because N is a nonzero digit.

Now, for divisibility by 3, I need to calculate the sum of all the digits of the number 21420N and check if that sum is divisible by 3. Let me write down the digits: 2, 1, 4, 2, 0, and then N.

Let me add them up. Let's compute the sum step by step.

Starting with 2 + 1 = 3. Then 3 + 4 = 7. Then 7 + 2 = 9. Then 9 + 0 = 9. Then adding N, the total sum is 9 + N.

So, the sum of the digits is 9 + N. For the number to be divisible by 3, 9 + N must be a multiple of 3.

Now, since 9 is already a multiple of 3 (because 9 divided by 3 is 3), adding N to it will result in a multiple of 3 if N is also a multiple of 3. Wait, let me think again.

If 9 is divisible by 3, then adding N to it will make the total sum divisible by 3 if N is divisible by 3. Because if you have a number divisible by 3 and add another number divisible by 3, the result is still divisible by 3. Conversely, if N is not divisible by 3, then the sum won't be.

But let me check with an example. Suppose N is 0: 9 + 0 = 9, which is divisible by 3. But N can't be 0 because the problem says nonzero. If N is 1: 9 + 1 = 10, which is not divisible by 3. N=2: 9 + 2 = 11, not divisible by 3. N=3: 9 + 3 = 12, which is divisible by 3. N=4: 13, not. N=5: 14, nope. N=6: 15, yes. N=7: 16, nope. N=8: 17, nope. N=9: 18, yes.

But remember, N has to be even because of divisibility by 2. So from the earlier list of possible N (2,4,6,8), which of these make 9 + N divisible by 3? Let's check:

For N=2: 9 + 2 = 11. 11 divided by 3 is 3 with a remainder of 2. Not divisible.

N=4: 9 + 4 = 13. 13 divided by 3 is 4 with remainder 1. Not divisible.

N=6: 9 + 6 = 15. 15 divided by 3 is 5. Perfect, divisible by 3.

N=8: 9 + 8 = 17. 17 divided by 3 is 5 with remainder 2. Not divisible.

So only N=6 satisfies both conditions: it's even (so divisible by 2) and 9 + 6 = 15 is divisible by 3. Therefore, N=6 is the answer.

Wait, but let me double-check. Let me write out the number 214206 and check if it's divisible by 6.

First, check divisibility by 2: the last digit is 6, which is even. Good.

Divisibility by 3: sum of digits is 2 + 1 + 4 + 2 + 0 + 6. Let me compute that again. 2 + 1 is 3, plus 4 is 7, plus 2 is 9, plus 0 is 9, plus 6 is 15. 15 is divisible by 3. So yes, 214206 is divisible by 3. Therefore, it's divisible by 6.

Is there another possible N? Let me check again. The possible Ns are 2,4,6,8. For N=6, it works. What about N=3? Wait, N=3 is not even, so it's excluded. Similarly, N=9 is not even. So only N=6.

But let me check if there's a mistake in my calculation. Maybe I miscalculated the sum of digits? Let me check again. The original number is 21420N. So digits are 2, 1, 4, 2, 0, N. So adding them up: 2 + 1 is 3, plus 4 is 7, plus 2 is 9, plus 0 is 9, plus N. So yes, 9 + N.

So 9 + N must be divisible by 3. Since 9 is already divisible by 3, N must be divisible by 3. But N is also even. So N must be a digit that is both even and divisible by 3. The digits that are even are 0,2,4,6,8. Among these, which are divisible by 3? Well, 0 is divisible by 3, but N is nonzero. So 6 is the only one. Because 6 divided by 3 is 2. 0 is excluded. So yes, N=6.

Therefore, the answer is 6.

But let me check if there's another way to approach this. Maybe by considering the entire number. Let me think.

The number is 21420N. Let me think of it as 214200 + N. Wait, no. Wait, 21420N is a 6-digit number? Wait, hold on. Wait, the original number is written as 21420N. Wait, is that 21420 followed by N? So if N is a single digit, then the number is 21420N, which would be a 6-digit number. For example, if N is 6, it's 214206.

But maybe the original problem is written as 21420N, where N is the units digit. So the number is 21420N, meaning that the original number is 21420 with N as the last digit. So yes, that's a 6-digit number.

Alternatively, maybe the problem is written as 21420N, where N is a digit, so the number is 21420N. So, for example, if N is 6, then the number is 214206.

But maybe I need to check if there's a different interpretation. But I think that's the correct way.

Alternatively, maybe the problem is written as 21420N, where N is a digit, so the entire number is 21420N. So, for example, if N is 6, then the number is 214206.

So, in that case, the number is 21420N.

But maybe I can check divisibility by 6 by using the actual number. Let me try with N=6.

214206 divided by 6. Let me compute that.

First, divide by 2: 214206 / 2 = 107103. Then divide by 3: 107103 / 3. Let me compute that.

Divide 107103 by 3: 3 goes into 10 three times (3*3=9), remainder 1. Bring down 7: 17. 3 into 17 is 5 times (15), remainder 2. Bring down 1: 21. 3 into 21 is 7. Bring down 0: 0. 3 into 0 is 0. Bring down 3: 3 into 3 is 1. So the result is 35701. So 3*35701=107103. Therefore, 214206 divided by 6 is 35701, which is an integer. Therefore, yes, it is divisible by 6.

Alternatively, if I tried N=2: 214202. Let me check divisibility by 6. Divisible by 2? Yes, ends with 2. Divisible by 3? Sum of digits: 2+1+4+2+0+2= 11. 11 is not divisible by 3. Therefore, not divisible by 3.

N=4: sum is 2+1+4+2+0+4= 13, not divisible by 3.

N=6: sum is 15, which is divisible by 3.

N=8: sum is 17, not divisible by 3.

So only N=6 works.

Therefore, the answer is 6.

But let me check once more if there's a mistake in my logic. For example, maybe I miscalculated the sum of digits? Let me check again.

Original number: 2, 1, 4, 2, 0, N.

Adding them up: 2 + 1 = 3; 3 + 4 = 7; 7 + 2 = 9; 9 + 0 = 9; 9 + N.

Yes, that's correct. So 9 + N.

So 9 + N must be divisible by 3. Since 9 is already divisible by 3, N must be divisible by 3.

But N is also even. So N must be even and divisible by 3. The digits that are even and divisible by 3 are 0 and 6. Since N is nonzero, only 6.

Therefore, N=6.

Therefore, the answer is 6.

I think that's solid. I don't see any mistakes in my reasoning. So I think the correct answer is 6.

**Final Answer**
The nonzero value of $ N $ that makes the number divisible by 6 is \boxed{6}.

To determine the nonzero value of $ N $ that makes the number $ 21420N $ divisible by 6, we need to apply the rules of divisibility by 6. A number is divisible by 6 if and only if it is divisible by both 2 and 3.

---

### Step 1: Divisibility by 2

A number is divisible by 2 if its **last digit is even**. In this case, the last digit is $ N $. Therefore, $ N $ must be an even digit.

The possible even digits are:
$$
N \in \{2, 4, 6, 8\}
$$

---

### Step 2: Divisibility by 3

A number is divisible by 3 if the **sum of its digits is divisible by 3**. The digits of the number $ 21420N $ are:
$$
2, 1, 4, 2, 0, N
$$

Adding these digits:
$$
2 + 1 + 4 + 2 + 0 + N = 9 + N
$$

So, the sum of the digits is $ 9 + N $. For the number to be divisible by 3, $ 9 + N $ must be divisible by 3.

Since 9 is already divisible by 3, $ N $ must also be divisible by 3 for the sum to remain divisible by 3.

---

### Step 3: Combine Both Conditions

From Step 1, $ N $ must be even:
$$
N \in \{2, 4, 6, 8\}
$$

From Step 2, $ N $ must be divisible by 3:
$$
N \in \{3, 6, 9\}
$$

The only digit that satisfies **both** conditions is:
$$
N = 6
$$

---

### Step 4: Final Verification

Let’s verify that $ N = 6 $ makes the number $ 214206 $ divisible by 6:

- **Divisibility by 2**: The last digit is 6, which is even ✅
- **Divisibility by 3**: Sum of digits = $ 2 + 1 + 4 + 2 + 0 + 6 = 15 $, and 15 is divisible by 3 ✅

Thus, the number is divisible by both 2 and 3, and therefore by 6.

---

### Final Answer

$$
\boxed{6}
$$

Solution: The number $21420N$ is divisible by $6$ if and only if it is even and the sum of its digits is divisible by $3$. So $N$ must be even, and $2 + 1 + 4 + 2 + 0 + N = 9 + N$ must be divisible by $3$.  Since $9$ is divisible by $3$, we see that $N$ must also be divisible by $3$. The only digit that works is $N = \boxed{6}$.


Explanation:
Okay, so I need to simplify this expression: (10r³)(4r⁶) divided by 8r⁴. Let me think about how to approach this.

First, I remember that when you multiply terms with the same base, you add the exponents. But here, there's also division involved. Maybe I should handle the coefficients first and then the variables? Let me try that.

Starting with the coefficients: 10 multiplied by 4, and then divided by 8. Let me calculate that. 10 times 4 is 40. Then, 40 divided by 8 is... 5. Okay, so the coefficient part simplifies to 5. That seems straightforward.

Now, the variables. The numerator has r³ multiplied by r⁶. When you multiply exponents with the same base, you add the exponents. So, r³ times r⁶ would be r^(3+6) which is r⁹. Then, the denominator is r⁴. So, dividing r⁹ by r⁴. For division, you subtract the exponents. So, r^(9-4) is r⁵.

Putting it all together, the coefficient is 5 and the variable part is r⁵. So, the simplified expression should be 5r⁵. Let me check if I did everything right.

Wait, let me verify each step again. First, coefficients: 10*4=40, then 40/8=5. Yep, that's correct. Then, for the variables: multiplying r³ and r⁶ gives r^(3+6)=r⁹. Then dividing by r⁴ is r^(9-4)=r⁵. So, combining them gives 5r⁵.

Hmm, seems right. Let me think if there's another way to approach this. Maybe simplifying before multiplying? Like, simplifying the coefficients first. Let me try that.

Original expression: (10r³)(4r⁶)/8r⁴. Maybe I can write this as (10/8)*(4/1)*(r³*r⁶)/r⁴. Wait, no, that might complicate. Alternatively, maybe factor out the numbers.

10 divided by 8 is 5/4, and then multiplied by 4? Let me see. Let me write it as (10*4)/(8) * r³*r⁶ / r⁴. Which is (40/8) * r^(3+6-4). Which is 5 * r^5. Yep, same result.

Alternatively, maybe simplifying fractions before multiplying. For example, 10 and 8 have a common factor of 2. So, 10 divided by 2 is 5, and 8 divided by 2 is 4. Then, 4 divided by 4 is 1. Wait, let me think.

Original coefficients: 10*4 /8. Let me write that as (10/8)*4. Which is (5/4)*4 = 5. Oh, that's another way to see it. So, 10/8 simplifies to 5/4, and then multiplied by 4 gives 5. So that works.

So, regardless of the method, the coefficient simplifies to 5. And the exponents for r: 3 + 6 - 4 = 5. So, r^5.

I think that's correct. Let me check with actual numbers. Let me plug in a value for r and see if both the original expression and the simplified one give the same result.

Let's choose r = 2.

Original expression: (10*(2)^3)*(4*(2)^6) divided by (8*(2)^4).

Calculating numerator: 10*8 = 80, 4*(2)^6 = 4*64 = 256. Multiply those: 80*256. Let me compute that. 80*256. Well, 80*200=16,000 and 80*56=4,480. So total is 16,000 + 4,480 = 20,480.

Denominator: 8*(2)^4 = 8*16 = 128.

So, 20,480 divided by 128. Let me compute that. 128*160 = 20,480. So, 20,480 / 128 = 160.

Now, simplified expression: 5*(2)^5. 2^5 is 32. 5*32 = 160.

Same result! So that checks out.

Let me try another number, maybe r = 1.

Original expression: (10*1^3)*(4*1^6)/ (8*1^4) = (10*1)*(4*1)/ (8*1) = 10*4 /8 = 40/8 = 5.

Simplified expression: 5*(1)^5 = 5*1 = 5. Same result.

Another test: r = 3.

Original expression: (10*27)*(4*729) / (8*81). Let me compute numerator: 10*27 = 270, 4*729 = 2916. Multiply them: 270*2916. Hmm, that's a big number. Let me see if I can compute it step by step.

Alternatively, maybe compute the simplified expression first. 5*(3)^5. 3^5 is 243. 5*243 = 1215.

Now, original expression: numerator is 270*2916. Let me compute denominator: 8*81 = 648.

So, 270*2916 / 648. Let me simplify this fraction.

First, note that 270 divided by 648. Let me see if I can divide numerator and denominator by something. Let me see, 270 and 648. Both divisible by 54? 270 ÷ 54 = 5, 648 ÷ 54 = 12. So, 5*2916 / 12.

Wait, 5*2916 divided by 12. Let me compute 2916 divided by 12 first. 12*243 = 2916. Because 12*200=2400, 12*43=516, so 2400+516=2916. So, 2916 /12 = 243. Then multiply by 5: 243*5=1215. Which matches the simplified expression.

So, that works too.

Therefore, my simplification seems correct.

Another thing to check: exponents. When multiplying r³ and r⁶, adding exponents is correct. Then dividing by r⁴, subtracting exponents. So 3+6=9, 9-4=5. Correct.

So, the simplified form is 5r⁵.

I think that's the answer. Let me just make sure there's no negative exponents or anything. The original problem didn't have any negative exponents, and since we're dividing by r⁴, as long as r ≠ 0, it's fine. But since the problem didn't specify, we can assume r is not zero.

So, yeah, I think that's it. The simplified expression is 5r⁵.

**Final Answer**
The simplified expression is \boxed{5r^5}.

To simplify the expression:

$$
\frac{(10r^3)(4r^6)}{8r^4}
$$

we will break the problem into two parts: simplifying the **coefficients** and simplifying the **variables** (i.e., the powers of $ r $).

---

### Step 1: Simplify the Coefficients

We first handle the numerical coefficients:

$$
\frac{10 \cdot 4}{8} = \frac{40}{8} = 5
$$

So, the coefficient simplifies to **5**.

---

### Step 2: Simplify the Variables

Now, we simplify the powers of $ r $. Recall the rules of exponents:

- **Multiplication**: $ r^a \cdot r^b = r^{a+b} $
- **Division**: $ \frac{r^a}{r^b} = r^{a-b} $

Apply these rules:

- Multiply $ r^3 $ and $ r^6 $:
  $$
  r^3 \cdot r^6 = r^{3+6} = r^9
  $$

- Divide by $ r^4 $:
  $$
  \frac{r^9}{r^4} = r^{9-4} = r^5
  $$

---

### Step 3: Combine the Results

Now, combine the simplified coefficient and variable:

$$
5 \cdot r^5 = 5r^5
$$

---

### Final Answer

$$
\boxed{5r^5}
$$

[Assistant]
Explanation:
Okay, so I need to simplify this expression: (10r³)(4r⁶) divided by 8r⁴. Let me think about how to approach this.

First, I remember that when you multiply terms with the same base, you add the exponents. But here, there's also division involved. Maybe I should handle the coefficients first and then the variables? Let me try that.

Starting with the coefficients: 10 multiplied by 4, and then divided by 8. Let me calculate that. 10 times 4 is 40. Then, 40 divided by 8 is... 5. Okay, so the coefficient part simplifies to 5. That seems straightforward.

Now, the variables. The numerator has r³ multiplied by r⁶. When you multiply exponents with the same base, you add the exponents. So, r³ times r⁶ would be r^(3+6) which is r⁹. Then, the denominator is r⁴. So, dividing r⁹ by r⁴. For division, you subtract the exponents. So, r^(9-4) is r⁵.

Putting it all together, the coefficient is 5 and the variable part is r⁵. So, the simplified expression should be 5r⁵. Let me check if I did everything right.

Wait, let me verify each step again. First, coefficients: 10*4=40, then 40/8=5. Yep, that's correct. Then, for the variables: multiplying r³ and r⁶ gives r^(3+6)=r⁹. Then dividing by r⁴ is r^(9-4)=r⁵. So, combining them gives 5r⁵.

Hmm, seems right. Let me think if there's another way to approach this. Maybe simplifying before multiplying? Like, simplifying the coefficients first. Let me try that.

Original expression: (10r³)(4r⁶)/8r⁴. Maybe I can write this as (10/8)*(4/1)*(r³*r⁶)/r⁴. Wait, no, that might complicate. Alternatively, maybe factor out the numbers.

10 divided by 8 is 5/4, and then multiplied by 4? Let me see. Let me write it as (10*4)/(8) * r³*r⁶ / r⁴. Which is (40/8) * r^(3+6-4). Which is 5 * r^5. Yep, same result.

Alternatively, maybe simplifying fractions before multiplying. For example, 10 and 8 have a common factor of 2. So, 10 divided by 2 is 5, and 8 divided by 2 is 4. Then, 4 divided by 4 is 1. Wait, let me think.

Original coefficients: 10*4 /8. Let me write that as (10/8)*4. Which is (5/4)*4 = 5. Oh, that's another way to see it. So, 10/8 simplifies to 5/4, and then multiplied by 4 gives 5. So that works.

So, regardless of the method, the coefficient simplifies to 5. And the exponents for r: 3 + 6 - 4 = 5. So, r^5.

I think that's correct. Let me check with actual numbers. Let me plug in a value for r and see if both the original expression and the simplified one give the same result.

Let's choose r = 2.

Original expression: (10*(2)^3)*(4*(2)^6) divided by (8*(2)^4).

Calculating numerator: 10*8 = 80, 4*(2)^6 = 4*64 = 256. Multiply those: 80*256. Let me compute that. 80*256. Well, 80*200=16,000 and 80*56=4,480. So total is 16,000 + 4,480 = 20,480.

Denominator: 8*(2)^4 = 8*16 = 128.

So, 20,480 divided by 128. Let me compute that. 128*160 = 20,480. So, 20,480 / 128 = 160.

Now, simplified expression: 5*(2)^5. 2^5 is 32. 5*32 = 160.

Same result! So that checks out.

Let me try another number, maybe r = 1.

Original expression: (10*1^3)*(4*1^6)/ (8*1^4) = (10*1)*(4*1)/ (8*1) = 10*4 /8 = 40/8 = 5.

Simplified expression: 5*(1)^5 = 5*1 = 5. Same result.

Another test: r = 3.

Original expression: (10*27)*(4*729) / (8*81). Let me compute numerator: 10*27 = 270, 4*729 = 2916. Multiply them: 270*2916. Hmm, that's a big number. Let me see if I can compute it step by step.

Alternatively, maybe compute the simplified expression first. 5*(3)^5. 3^5 is 243. 5*243 = 1215.

Now, original expression: numerator is 270*2916. Let me compute denominator: 8*81 = 648.

So, 270*2916 / 648. Let me simplify this fraction.

First, note that 270 divided by 648. Let me see if I can divide numerator and denominator by something. Let me see, 270 and 648. Both divisible by 54? 270 ÷ 54 = 5, 648 ÷ 54 = 12. So, 5*2916 / 12.

Wait, 5*2916 divided by 12. Let me compute 2916 divided by 12 first. 12*243 = 2916. Because 12*200=2400, 12*43=516, so 2400+516=2916. So, 2916 /12 = 243. Then multiply by 5: 243*5=1215. Which matches the simplified expression.

So, that works too.

Therefore, my simplification seems correct.

Another thing to check: exponents. When multiplying r³ and r⁶, adding exponents is correct. Then dividing by r⁴, subtracting exponents. So 3+6=9, 9-4=5. Correct.

So, the simplified form is 5r⁵.

I think that's the answer. Let me just make sure there's no negative exponents or anything. The original problem didn't have any negative exponents, and since we're dividing by r⁴, as long as r ≠ 0, it's fine. But since the problem didn't specify, we can assume r is not zero.

So, yeah, I think that's it. The simplified expression is 5r⁵.

**Final Answer**
The simplified expression is \boxed{5r^5}.

To simplify the expression:

$$
\frac{(10r^3)(4r^6)}{8r^4}
$$

we will break the problem into two parts: simplifying the **coefficients** and simplifying the **variables** (i.e., the powers of $ r $).

---

### Step 1: Simplify the Coefficients

We first handle the numerical coefficients:

$$
\frac{10 \cdot 4}{8} = \frac{40}{8} = 5
$$

So, the coefficient simplifies to **5**.

---

### Step 2: Simplify the Variables

Now, we simplify the powers of $ r $. Recall the rules of exponents:

- **Multiplication**: $ r^a \cdot r^b = r^{a+b} $
- **Division**: $ \frac{r^a}{r^b} = r^{a-b} $

Apply these rules:

- Multiply $ r^3 $ and $ r^6 $:
  $$
  r^3 \cdot r^6 = r^{3+6} = r^9
  $$

- Divide by $ r^4 $:
  $$
  \frac{r^9}{r^4} = r^{9-4} = r^5
  $$

---

### Step 3: Combine the Results

Now, combine the simplified coefficient and variable:

$$
5 \cdot r^5 = 5r^5
$$

---

### Final Answer

$$
\boxed{5r^5}
$$

Solution: We have  \[\frac{(10r^3)(4r^6)}{8r^4}= \frac{40r^{3+6}}{8r^4} = \frac{40}{8}r^{3+6-4} = \boxed{5r^5}.\]