Explanation:
Okay, so I need to find the equation of the plane that contains the three points given: (0, -1, -1), (-4, 4, 4), and (4, 5, 1). The answer should be in the form Ax + By + Cz + D = 0, with A, B, C, D as integers, A positive, and the greatest common divisor of their absolute values being 1. Alright, let me recall how to do this.

First, I remember that to find the equation of a plane, you need a point on the plane and a normal vector to the plane. Since we have three points, we can use them to find two vectors lying on the plane, then take their cross product to get the normal vector. Once we have the normal vector, we can plug it into the plane equation along with one of the points to solve for D.

Let me write down the points again for clarity:

Point P: (0, -1, -1)
Point Q: (-4, 4, 4)
Point R: (4, 5, 1)

So, first step: find two vectors on the plane. Let me subtract coordinates of the points to get vectors PQ and PR or something like that.

Let me compute vector PQ and PR. Wait, actually, vectors can be found by subtracting coordinates of two points. Let me pick vectors PQ and PR. Wait, but maybe it's better to use vectors from the same point. Let me think.

Alternatively, I can take vectors PQ and QR or PR and QR. Let me just pick two vectors. Let me take vectors from point P to Q and from point P to R. That should work.

So vector PQ is Q - P. Let me compute that:

Q - P = (-4 - 0, 4 - (-1), 4 - (-1)) = (-4, 5, 5)

Similarly, vector PR is R - P = (4 - 0, 5 - (-1), 1 - (-1)) = (4, 6, 2)

Okay, so now I have two vectors in the plane: PQ = (-4, 5, 5) and PR = (4, 6, 2). Now, the cross product of these two vectors will give me a normal vector to the plane.

Let me recall how to compute the cross product. If I have two vectors, say vector a = (a1, a2, a3) and vector b = (b1, b2, b3), then their cross product is:

(a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)

Alternatively, using the determinant formula with unit vectors i, j, k.

Let me write that out for PQ and PR.

Let me denote vector PQ as a = (-4, 5, 5) and vector PR as b = (4, 6, 2). Then cross product a × b is:

|i   j   k |
|-4 5   5|
|4   6   2|

Calculating determinant:

i*(5*2 - 5*6) - j*(-4*2 - 5*4) + k*(-4*6 - 5*4)

Let me compute each component step by step.

First component (i):

5*2 - 5*6 = 10 - 30 = -20

Second component (j):

- [ (-4)*2 - 5*4 ] = - [ -8 - 20 ] = - [ -28 ] = 28

Wait, hold on. Wait, the formula is:

For the cross product, the j component is subtracted, so it's - (a1b3 - a3b1). Wait, maybe I should double-check.

Alternatively, using the standard formula:

If a = (a1, a2, a3), b = (b1, b2, b3), then cross product is:

(a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)

So for the first component (x-component): a2b3 - a3b2 = 5*2 - 5*6 = 10 - 30 = -20

Second component (y-component): a3b1 - a1b3 = 5*4 - (-4)*2 = 20 - (-8) = 20 + 8 = 28

Third component (z-component): a1b2 - a2b1 = (-4)*6 - 5*4 = -24 - 20 = -44

Therefore, the cross product vector is (-20, 28, -44). So this is the normal vector n = (-20, 28, -44).

But we can simplify this vector if possible. Let me check if there is a common factor. Let's see:

The components are -20, 28, -44. Let me find the greatest common divisor of 20, 28, 44.

Factors of 20: 2, 4, 5, 10, 20

Factors of 28: 2, 4, 7, 14, 28

Factors of 44: 2, 4, 11, 22, 44

Common factors are 2 and 4. Let me check if 4 divides all:

-20 divided by 4 is -5, 28 divided by 4 is 7, -44 divided by 4 is -11. So yes, we can divide by 4. So the normal vector can be simplified to (-5, 7, -11). Let me check:

Original cross product: (-20, 28, -44). Divided by 4: (-5, 7, -11). Yes. So that's a simpler normal vector.

Alternatively, since the direction of the normal vector doesn't matter (as long as it's perpendicular), but we can also take the opposite direction if needed. However, since we need A > 0 in the plane equation, we need to make sure that the coefficient A is positive. Let me check the normal vector (-5, 7, -11). The first component is -5, which is negative. So if we take the normal vector as (-5, 7, -11), then in the plane equation, A would be -5, but we need A > 0. Therefore, we can multiply the entire normal vector by -1 to get (5, -7, 11). That way, A is positive. So the normal vector is (5, -7, 11). Let me confirm that.

Yes, because multiplying by -1 reverses the direction, but it's still a normal vector. So that's better for our purposes.

So now, the normal vector is (5, -7, 11). Now, we can use this normal vector and one of the points to write the plane equation.

The general equation of a plane is A(x - x0) + B(y - y0) + C(z - z0) = 0, where (x0, y0, z0) is a point on the plane, and (A, B, C) is the normal vector.

Let me pick point P (0, -1, -1) as the point on the plane. So plugging into the equation:

5(x - 0) + (-7)(y - (-1)) + 11(z - (-1)) = 0

Simplify:

5x -7(y + 1) + 11(z + 1) = 0

Expanding:

5x -7y -7 + 11z + 11 = 0

Combine constants: -7 + 11 = 4

So:

5x -7y + 11z + 4 = 0

Wait, so the equation is 5x -7y +11z +4 = 0. Let me check if this is correct. Let me verify if all three points satisfy this equation.

First, check point P: (0, -1, -1)

Plug in: 5*0 -7*(-1) +11*(-1) +4 = 0 +7 -11 +4 = 0. 7 -11 is -4, +4 is 0. Correct.

Check point Q: (-4, 4, 4)

Plug in: 5*(-4) -7*(4) +11*(4) +4 = -20 -28 +44 +4 = (-20 -28) + (44 +4) = -48 + 48 = 0. Correct.

Check point R: (4, 5, 1)

Plug in: 5*4 -7*5 +11*1 +4 = 20 -35 +11 +4 = (20 -35) + (11 +4) = (-15) + 15 = 0. Correct. Perfect, all three points satisfy the equation. So that seems right.

But let me check if the coefficients have a common divisor. The coefficients are 5, -7, 11, and 4. Let me check gcd of 5, 7, 11, 4. Well, 5 is prime, 7 is prime, 11 is prime, and 4 is 2^2. The gcd of all these numbers is 1, since they don't share any common factors besides 1. Therefore, the equation 5x -7y +11z +4 = 0 satisfies the condition that gcd(|A|, |B|, |C|, |D|) = 1, and A is positive. So that's the answer?

Wait, but let me check again. The problem says "Enter your answer in the form Ax + By + Cz + D = 0, where A, B, C, D are integers such that A > 0 and gcd(|A|, |B|, |C|, |D|) = 1."

So our equation is 5x -7y +11z +4 = 0. Let me check the gcd of 5, 7, 11, 4. The absolute values are 5,7,11,4. The gcd of these numbers is 1, since 5 is prime, 7 is prime, 11 is prime, and 4 is 2^2. There is no common divisor other than 1. So that's good. Also, A is 5, which is positive. So that seems to satisfy all the conditions.

But let me just make sure that I didn't make any mistakes in the cross product or the calculation. Let me recheck the cross product step.

Original vectors PQ = (-4, 5, 5) and PR = (4, 6, 2). Cross product:

i*(5*2 - 5*6) - j*(-4*2 - 5*4) + k*(-4*6 - 5*4)

Wait, earlier I calculated the cross product as (-20, 28, -44). Let me recompute:

First component (i): (5*2 - 5*6) = 10 - 30 = -20. Correct.

Second component (j): -( (-4*2 - 5*4) ) = - ( -8 -20 ) = -(-28) = 28. Correct.

Third component (k): (-4*6 - 5*4) = -24 -20 = -44. Correct. So cross product is (-20, 28, -44). Then dividing by 4 gives (-5, 7, -11). Then multiplying by -1 gives (5, -7, 11). Correct.

Then using point P (0, -1, -1):

5(x - 0) -7(y - (-1)) + 11(z - (-1)) = 0 => 5x -7(y +1) +11(z +1) = 0 => 5x -7y -7 +11z +11 = 0 => 5x -7y +11z +4 = 0. Correct.

So that seems correct. Therefore, the answer is 5x -7y +11z +4 = 0. Let me check once more if there's a possible mistake in signs or coefficients.

Alternatively, maybe I can use another point to derive the equation. Let me try using point Q (-4, 4, 4). Let me use the normal vector (5, -7, 11). The plane equation is 5x -7y +11z + D = 0. Plug in point Q:

5*(-4) -7*(4) +11*(4) + D = 0 => -20 -28 +44 + D = 0 => (-48 + 44) + D = 0 => -4 + D = 0 => D = 4. So same result. Similarly, using point R (4,5,1):

5*4 -7*5 +11*1 + D = 20 -35 +11 + D = (-15 +11) + D = -4 + D = 0 => D = 4. So same D. So that's consistent. Therefore, the equation is correct.

Therefore, the answer is 5x -7y +11z +4 = 0. So in the required form, that's Ax + By + Cz + D = 0 with A=5, B=-7, C=11, D=4. Since they want A>0, which is satisfied, and gcd(5,7,11,4)=1. So that's the answer.

But just to be thorough, let me check if there's another way to approach this problem, maybe using the general plane equation and solving the system.

The general equation of a plane is Ax + By + Cz + D = 0. Since the three points lie on the plane, plugging them in gives three equations:

For point P (0, -1, -1):

A*0 + B*(-1) + C*(-1) + D = 0 => -B - C + D = 0 => -B - C + D = 0. Equation (1)

For point Q (-4, 4, 4):

A*(-4) + B*4 + C*4 + D = 0 => -4A +4B +4C + D = 0. Equation (2)

For point R (4, 5, 1):

A*4 + B*5 + C*1 + D = 0 => 4A +5B + C + D = 0. Equation (3)

So now, we have three equations:

1) -B - C + D = 0

2) -4A +4B +4C + D = 0

3) 4A +5B + C + D = 0

We need to solve for A, B, C, D. However, since the plane equation is determined up to a scalar multiple, we can set one variable to a value (like A=5 as before) and solve, but let me try solving this system.

Let me write equations:

From equation (1): D = B + C.

Let me substitute D = B + C into equations (2) and (3):

Equation (2): -4A +4B +4C + (B + C) = 0 => -4A +5B +5C = 0 => -4A +5B +5C = 0. Equation (2a)

Equation (3): 4A +5B + C + (B + C) = 0 => 4A +5B + C + B + C = 0 => 4A +6B + 2C = 0. Equation (3a)

So now, equations (2a) and (3a):

(2a): -4A +5B +5C = 0

(3a): 4A +6B +2C = 0

Let me add equations (2a) and (3a) to eliminate A:

(-4A +4A) + (5B +6B) + (5C +2C) = 0 + 0 => 0 + 11B +7C = 0 => 11B +7C = 0. Equation (4)

So from equation (4): 11B +7C = 0 => 11B = -7C => B = (-7/11)C.

Let me express B in terms of C. Let me choose C as a parameter. Let me let C = 11k, then B = -7k, where k is some constant. Let me choose k = 1 for simplicity, so C = 11, B = -7. Then, from equation (1): D = B + C = -7 + 11 = 4. So D = 4.

Now, substitute B = -7, C = 11 into equation (2a) or (3a) to find A.

Let me use equation (2a): -4A +5B +5C = 0 => -4A +5*(-7) +5*(11) = 0 => -4A -35 +55 = 0 => -4A +20 = 0 => -4A = -20 => A = 5. Perfect! So A = 5, B = -7, C = 11, D = 4. Which is exactly the same result as before. Therefore, this confirms that the equation is indeed 5x -7y +11z +4 = 0. So that's correct.

Therefore, the answer is 5x -7y +11z +4 = 0. Let me check once more if the gcd is 1. The coefficients are 5, -7, 11, 4. The absolute values are 5,7,11,4. The gcd of these numbers is 1, since 5 is prime, 7 is prime, 11 is prime, and 4 is 2^2. There is no common divisor other than 1. Therefore, this satisfies the condition. Also, A is positive. So this is the correct answer.

I think that's thorough. I considered two methods: cross product and solving the system of equations, both leading to the same result. Verified all three points satisfy the equation, checked the gcd condition. Seems solid.

**Final Answer**
The equation of the plane is \boxed{5x - 7y + 11z + 4 = 0}.

To find the equation of the plane containing the points $ P(0, -1, -1) $, $ Q(-4, 4, 4) $, and $ R(4, 5, 1) $, we proceed as follows:

---

### **Step 1: Find Two Vectors in the Plane**

We compute two vectors that lie on the plane by subtracting coordinates of the given points:

- $ \vec{PQ} = Q - P = (-4 - 0, 4 - (-1), 4 - (-1)) = (-4, 5, 5) $
- $ \vec{PR} = R - P = (4 - 0, 5 - (-1), 1 - (-1)) = (4, 6, 2) $

---

### **Step 2: Compute the Normal Vector via Cross Product**

The cross product of $ \vec{PQ} $ and $ \vec{PR} $ gives a vector normal to the plane:

$$
\vec{n} = \vec{PQ} \times \vec{PR} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
-4 & 5 & 5 \\
4 & 6 & 2
\end{vmatrix}
= \mathbf{i}(5 \cdot 2 - 5 \cdot 6) - \mathbf{j}(-4 \cdot 2 - 5 \cdot 4) + \mathbf{k}(-4 \cdot 6 - 5 \cdot 4)
$$

$$
= \mathbf{i}(-20) - \mathbf{j}(-28) + \mathbf{k}(-44) = (-20, 28, -44)
$$

We simplify this vector by dividing by the greatest common divisor of its components (which is 4):

$$
\vec{n} = (-5, 7, -11)
$$

To ensure the coefficient of $ x $ is positive, we multiply by $ -1 $:

$$
\vec{n} = (5, -7, 11)
$$

---

### **Step 3: Use the Point-Normal Form of the Plane Equation**

Using the point $ P(0, -1, -1) $ and the normal vector $ (5, -7, 11) $, the plane equation is:

$$
5(x - 0) - 7(y + 1) + 11(z + 1) = 0
$$

Simplify:

$$
5x - 7y - 7 + 11z + 11 = 0 \Rightarrow 5x - 7y + 11z + 4 = 0
$$

---

### **Step 4: Verify the Equation**

We verify that all three points satisfy the equation:

- For $ P(0, -1, -1) $: $ 5(0) - 7(-1) + 11(-1) + 4 = 0 + 7 - 11 + 4 = 0 $
- For $ Q(-4, 4, 4) $: $ 5(-4) - 7(4) + 11(4) + 4 = -20 - 28 + 44 + 4 = 0 $
- For $ R(4, 5, 1) $: $ 5(4) - 7(5) + 11(1) + 4 = 20 - 35 + 11 + 4 = 0 $

All points satisfy the equation.

---

### **Step 5: Ensure the Form and Conditions Are Met**

The equation is:

$$
5x - 7y + 11z + 4 = 0
$$

- $ A = 5 > 0 $
- The greatest common divisor of $ |5|, |7|, |11|, |4| $ is 1

Thus, the equation satisfies all the required conditions.

---

### **Final Answer**

$$
\boxed{5x - 7y + 11z + 4 = 0}
$$

[Assistant]
Explanation:
Okay, so I need to find the equation of the plane that contains the three points given: (0, -1, -1), (-4, 4, 4), and (4, 5, 1). The answer should be in the form Ax + By + Cz + D = 0, with A, B, C, D as integers, A positive, and the greatest common divisor of their absolute values being 1. Alright, let me recall how to do this.

First, I remember that to find the equation of a plane, you need a point on the plane and a normal vector to the plane. Since we have three points, we can use them to find two vectors lying on the plane, then take their cross product to get the normal vector. Once we have the normal vector, we can plug it into the plane equation along with one of the points to solve for D.

Let me write down the points again for clarity:

Point P: (0, -1, -1)
Point Q: (-4, 4, 4)
Point R: (4, 5, 1)

So, first step: find two vectors on the plane. Let me subtract coordinates of the points to get vectors PQ and PR or something like that.

Let me compute vector PQ and PR. Wait, actually, vectors can be found by subtracting coordinates of two points. Let me pick vectors PQ and PR. Wait, but maybe it's better to use vectors from the same point. Let me think.

Alternatively, I can take vectors PQ and QR or PR and QR. Let me just pick two vectors. Let me take vectors from point P to Q and from point P to R. That should work.

So vector PQ is Q - P. Let me compute that:

Q - P = (-4 - 0, 4 - (-1), 4 - (-1)) = (-4, 5, 5)

Similarly, vector PR is R - P = (4 - 0, 5 - (-1), 1 - (-1)) = (4, 6, 2)

Okay, so now I have two vectors in the plane: PQ = (-4, 5, 5) and PR = (4, 6, 2). Now, the cross product of these two vectors will give me a normal vector to the plane.

Let me recall how to compute the cross product. If I have two vectors, say vector a = (a1, a2, a3) and vector b = (b1, b2, b3), then their cross product is:

(a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)

Alternatively, using the determinant formula with unit vectors i, j, k.

Let me write that out for PQ and PR.

Let me denote vector PQ as a = (-4, 5, 5) and vector PR as b = (4, 6, 2). Then cross product a × b is:

|i   j   k |
|-4 5   5|
|4   6   2|

Calculating determinant:

i*(5*2 - 5*6) - j*(-4*2 - 5*4) + k*(-4*6 - 5*4)

Let me compute each component step by step.

First component (i):

5*2 - 5*6 = 10 - 30 = -20

Second component (j):

- [ (-4)*2 - 5*4 ] = - [ -8 - 20 ] = - [ -28 ] = 28

Wait, hold on. Wait, the formula is:

For the cross product, the j component is subtracted, so it's - (a1b3 - a3b1). Wait, maybe I should double-check.

Alternatively, using the standard formula:

If a = (a1, a2, a3), b = (b1, b2, b3), then cross product is:

(a2b3 - a3b2, a3b1 - a1b3, a1b2 - a2b1)

So for the first component (x-component): a2b3 - a3b2 = 5*2 - 5*6 = 10 - 30 = -20

Second component (y-component): a3b1 - a1b3 = 5*4 - (-4)*2 = 20 - (-8) = 20 + 8 = 28

Third component (z-component): a1b2 - a2b1 = (-4)*6 - 5*4 = -24 - 20 = -44

Therefore, the cross product vector is (-20, 28, -44). So this is the normal vector n = (-20, 28, -44).

But we can simplify this vector if possible. Let me check if there is a common factor. Let's see:

The components are -20, 28, -44. Let me find the greatest common divisor of 20, 28, 44.

Factors of 20: 2, 4, 5, 10, 20

Factors of 28: 2, 4, 7, 14, 28

Factors of 44: 2, 4, 11, 22, 44

Common factors are 2 and 4. Let me check if 4 divides all:

-20 divided by 4 is -5, 28 divided by 4 is 7, -44 divided by 4 is -11. So yes, we can divide by 4. So the normal vector can be simplified to (-5, 7, -11). Let me check:

Original cross product: (-20, 28, -44). Divided by 4: (-5, 7, -11). Yes. So that's a simpler normal vector.

Alternatively, since the direction of the normal vector doesn't matter (as long as it's perpendicular), but we can also take the opposite direction if needed. However, since we need A > 0 in the plane equation, we need to make sure that the coefficient A is positive. Let me check the normal vector (-5, 7, -11). The first component is -5, which is negative. So if we take the normal vector as (-5, 7, -11), then in the plane equation, A would be -5, but we need A > 0. Therefore, we can multiply the entire normal vector by -1 to get (5, -7, 11). That way, A is positive. So the normal vector is (5, -7, 11). Let me confirm that.

Yes, because multiplying by -1 reverses the direction, but it's still a normal vector. So that's better for our purposes.

So now, the normal vector is (5, -7, 11). Now, we can use this normal vector and one of the points to write the plane equation.

The general equation of a plane is A(x - x0) + B(y - y0) + C(z - z0) = 0, where (x0, y0, z0) is a point on the plane, and (A, B, C) is the normal vector.

Let me pick point P (0, -1, -1) as the point on the plane. So plugging into the equation:

5(x - 0) + (-7)(y - (-1)) + 11(z - (-1)) = 0

Simplify:

5x -7(y + 1) + 11(z + 1) = 0

Expanding:

5x -7y -7 + 11z + 11 = 0

Combine constants: -7 + 11 = 4

So:

5x -7y + 11z + 4 = 0

Wait, so the equation is 5x -7y +11z +4 = 0. Let me check if this is correct. Let me verify if all three points satisfy this equation.

First, check point P: (0, -1, -1)

Plug in: 5*0 -7*(-1) +11*(-1) +4 = 0 +7 -11 +4 = 0. 7 -11 is -4, +4 is 0. Correct.

Check point Q: (-4, 4, 4)

Plug in: 5*(-4) -7*(4) +11*(4) +4 = -20 -28 +44 +4 = (-20 -28) + (44 +4) = -48 + 48 = 0. Correct.

Check point R: (4, 5, 1)

Plug in: 5*4 -7*5 +11*1 +4 = 20 -35 +11 +4 = (20 -35) + (11 +4) = (-15) + 15 = 0. Correct. Perfect, all three points satisfy the equation. So that seems right.

But let me check if the coefficients have a common divisor. The coefficients are 5, -7, 11, and 4. Let me check gcd of 5, 7, 11, 4. Well, 5 is prime, 7 is prime, 11 is prime, and 4 is 2^2. The gcd of all these numbers is 1, since they don't share any common factors besides 1. Therefore, the equation 5x -7y +11z +4 = 0 satisfies the condition that gcd(|A|, |B|, |C|, |D|) = 1, and A is positive. So that's the answer?

Wait, but let me check again. The problem says "Enter your answer in the form Ax + By + Cz + D = 0, where A, B, C, D are integers such that A > 0 and gcd(|A|, |B|, |C|, |D|) = 1."

So our equation is 5x -7y +11z +4 = 0. Let me check the gcd of 5, 7, 11, 4. The absolute values are 5,7,11,4. The gcd of these numbers is 1, since 5 is prime, 7 is prime, 11 is prime, and 4 is 2^2. There is no common divisor other than 1. So that's good. Also, A is 5, which is positive. So that seems to satisfy all the conditions.

But let me just make sure that I didn't make any mistakes in the cross product or the calculation. Let me recheck the cross product step.

Original vectors PQ = (-4, 5, 5) and PR = (4, 6, 2). Cross product:

i*(5*2 - 5*6) - j*(-4*2 - 5*4) + k*(-4*6 - 5*4)

Wait, earlier I calculated the cross product as (-20, 28, -44). Let me recompute:

First component (i): (5*2 - 5*6) = 10 - 30 = -20. Correct.

Second component (j): -( (-4*2 - 5*4) ) = - ( -8 -20 ) = -(-28) = 28. Correct.

Third component (k): (-4*6 - 5*4) = -24 -20 = -44. Correct. So cross product is (-20, 28, -44). Then dividing by 4 gives (-5, 7, -11). Then multiplying by -1 gives (5, -7, 11). Correct.

Then using point P (0, -1, -1):

5(x - 0) -7(y - (-1)) + 11(z - (-1)) = 0 => 5x -7(y +1) +11(z +1) = 0 => 5x -7y -7 +11z +11 = 0 => 5x -7y +11z +4 = 0. Correct.

So that seems correct. Therefore, the answer is 5x -7y +11z +4 = 0. Let me check once more if there's a possible mistake in signs or coefficients.

Alternatively, maybe I can use another point to derive the equation. Let me try using point Q (-4, 4, 4). Let me use the normal vector (5, -7, 11). The plane equation is 5x -7y +11z + D = 0. Plug in point Q:

5*(-4) -7*(4) +11*(4) + D = 0 => -20 -28 +44 + D = 0 => (-48 + 44) + D = 0 => -4 + D = 0 => D = 4. So same result. Similarly, using point R (4,5,1):

5*4 -7*5 +11*1 + D = 20 -35 +11 + D = (-15 +11) + D = -4 + D = 0 => D = 4. So same D. So that's consistent. Therefore, the equation is correct.

Therefore, the answer is 5x -7y +11z +4 = 0. So in the required form, that's Ax + By + Cz + D = 0 with A=5, B=-7, C=11, D=4. Since they want A>0, which is satisfied, and gcd(5,7,11,4)=1. So that's the answer.

But just to be thorough, let me check if there's another way to approach this problem, maybe using the general plane equation and solving the system.

The general equation of a plane is Ax + By + Cz + D = 0. Since the three points lie on the plane, plugging them in gives three equations:

For point P (0, -1, -1):

A*0 + B*(-1) + C*(-1) + D = 0 => -B - C + D = 0 => -B - C + D = 0. Equation (1)

For point Q (-4, 4, 4):

A*(-4) + B*4 + C*4 + D = 0 => -4A +4B +4C + D = 0. Equation (2)

For point R (4, 5, 1):

A*4 + B*5 + C*1 + D = 0 => 4A +5B + C + D = 0. Equation (3)

So now, we have three equations:

1) -B - C + D = 0

2) -4A +4B +4C + D = 0

3) 4A +5B + C + D = 0

We need to solve for A, B, C, D. However, since the plane equation is determined up to a scalar multiple, we can set one variable to a value (like A=5 as before) and solve, but let me try solving this system.

Let me write equations:

From equation (1): D = B + C.

Let me substitute D = B + C into equations (2) and (3):

Equation (2): -4A +4B +4C + (B + C) = 0 => -4A +5B +5C = 0 => -4A +5B +5C = 0. Equation (2a)

Equation (3): 4A +5B + C + (B + C) = 0 => 4A +5B + C + B + C = 0 => 4A +6B + 2C = 0. Equation (3a)

So now, equations (2a) and (3a):

(2a): -4A +5B +5C = 0

(3a): 4A +6B +2C = 0

Let me add equations (2a) and (3a) to eliminate A:

(-4A +4A) + (5B +6B) + (5C +2C) = 0 + 0 => 0 + 11B +7C = 0 => 11B +7C = 0. Equation (4)

So from equation (4): 11B +7C = 0 => 11B = -7C => B = (-7/11)C.

Let me express B in terms of C. Let me choose C as a parameter. Let me let C = 11k, then B = -7k, where k is some constant. Let me choose k = 1 for simplicity, so C = 11, B = -7. Then, from equation (1): D = B + C = -7 + 11 = 4. So D = 4.

Now, substitute B = -7, C = 11 into equation (2a) or (3a) to find A.

Let me use equation (2a): -4A +5B +5C = 0 => -4A +5*(-7) +5*(11) = 0 => -4A -35 +55 = 0 => -4A +20 = 0 => -4A = -20 => A = 5. Perfect! So A = 5, B = -7, C = 11, D = 4. Which is exactly the same result as before. Therefore, this confirms that the equation is indeed 5x -7y +11z +4 = 0. So that's correct.

Therefore, the answer is 5x -7y +11z +4 = 0. Let me check once more if the gcd is 1. The coefficients are 5, -7, 11, 4. The absolute values are 5,7,11,4. The gcd of these numbers is 1, since 5 is prime, 7 is prime, 11 is prime, and 4 is 2^2. There is no common divisor other than 1. Therefore, this satisfies the condition. Also, A is positive. So this is the correct answer.

I think that's thorough. I considered two methods: cross product and solving the system of equations, both leading to the same result. Verified all three points satisfy the equation, checked the gcd condition. Seems solid.

**Final Answer**
The equation of the plane is \boxed{5x - 7y + 11z + 4 = 0}.

To find the equation of the plane containing the points $ P(0, -1, -1) $, $ Q(-4, 4, 4) $, and $ R(4, 5, 1) $, we proceed as follows:

---

### **Step 1: Find Two Vectors in the Plane**

We compute two vectors that lie on the plane by subtracting coordinates of the given points:

- $ \vec{PQ} = Q - P = (-4 - 0, 4 - (-1), 4 - (-1)) = (-4, 5, 5) $
- $ \vec{PR} = R - P = (4 - 0, 5 - (-1), 1 - (-1)) = (4, 6, 2) $

---

### **Step 2: Compute the Normal Vector via Cross Product**

The cross product of $ \vec{PQ} $ and $ \vec{PR} $ gives a vector normal to the plane:

$$
\vec{n} = \vec{PQ} \times \vec{PR} =
\begin{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
-4 & 5 & 5 \\
4 & 6 & 2
\end{vmatrix}
= \mathbf{i}(5 \cdot 2 - 5 \cdot 6) - \mathbf{j}(-4 \cdot 2 - 5 \cdot 4) + \mathbf{k}(-4 \cdot 6 - 5 \cdot 4)
$$

$$
= \mathbf{i}(-20) - \mathbf{j}(-28) + \mathbf{k}(-44) = (-20, 28, -44)
$$

We simplify this vector by dividing by the greatest common divisor of its components (which is 4):

$$
\vec{n} = (-5, 7, -11)
$$

To ensure the coefficient of $ x $ is positive, we multiply by $ -1 $:

$$
\vec{n} = (5, -7, 11)
$$

---

### **Step 3: Use the Point-Normal Form of the Plane Equation**

Using the point $ P(0, -1, -1) $ and the normal vector $ (5, -7, 11) $, the plane equation is:

$$
5(x - 0) - 7(y + 1) + 11(z + 1) = 0
$$

Simplify:

$$
5x - 7y - 7 + 11z + 11 = 0 \Rightarrow 5x - 7y + 11z + 4 = 0
$$

---

### **Step 4: Verify the Equation**

We verify that all three points satisfy the equation:

- For $ P(0, -1, -1) $: $ 5(0) - 7(-1) + 11(-1) + 4 = 0 + 7 - 11 + 4 = 0 $
- For $ Q(-4, 4, 4) $: $ 5(-4) - 7(4) + 11(4) + 4 = -20 - 28 + 44 + 4 = 0 $
- For $ R(4, 5, 1) $: $ 5(4) - 7(5) + 11(1) + 4 = 20 - 35 + 11 + 4 = 0 $

All points satisfy the equation.

---

### **Step 5: Ensure the Form and Conditions Are Met**

The equation is:

$$
5x - 7y + 11z + 4 = 0
$$

- $ A = 5 > 0 $
- The greatest common divisor of $ |5|, |7|, |11|, |4| $ is 1

Thus, the equation satisfies all the required conditions.

---

### **Final Answer**

$$
\boxed{5x - 7y + 11z + 4 = 0}
$$

Solution:
Let $\mathbf{a} = \begin{pmatrix} 0 \\ -1 \\ -1 \end{pmatrix},$ $\mathbf{b} = \begin{pmatrix} -4 \\ 4 \\ 4 \end{pmatrix},$ and $\mathbf{c} = \begin{pmatrix} 4 \\ 5 \\ 1 \end{pmatrix}.$  Then the normal vector of the plane is orthogonal to both
\[\mathbf{b} - \mathbf{a} = \begin{pmatrix} -4 \\ 5 \\ 5 \end{pmatrix}\]and
\[\mathbf{c} - \mathbf{a} = \begin{pmatrix} 4 \\ 6 \\ 2 \end{pmatrix}.\]So to compute the normal vector, we take the cross product of these vectors:
\[\begin{pmatrix} -4 \\ 5 \\ 5 \end{pmatrix} \times \begin{pmatrix} 4 \\ 6 \\ 2 \end{pmatrix} = \begin{pmatrix} -20 \\ 28 \\ -44 \end{pmatrix}.\]We can scale this vector, and take $\begin{pmatrix} 5 \\ -7 \\ 11 \end{pmatrix}$ as the normal vector.  Then the equation of the plane is of the form
\[5x - 7y + 11z + D = 0.\]Substituting the coordinates of any of the points, we find that the equation of the plane is $\boxed{5x - 7y + 11z + 4 = 0}.$


Explanation:
Okay, so I need to figure out how to write the expression √2 + 1/√2 + √3 + 1/√3 in the form (a√2 + b√3)/c, where a, b, and c are positive integers, and c is as small as possible. Then, I have to find a + b + c. Alright, let me start by breaking down the problem step by step.

First, let me look at each term individually. The expression has four terms: √2, 1/√2, √3, and 1/√3. Maybe I can combine the terms with √2 and the terms with √3 separately. That might make it easier to handle.

Starting with the √2 terms: √2 and 1/√2. Let me combine these two. To add them together, they need a common denominator. The term √2 can be written as √2/1, and 1/√2 is already a fraction. To add them, I can rationalize the denominator of 1/√2. Wait, but maybe I can just combine them directly?

Alternatively, maybe I can combine them by converting them to have the same denominator. Let me try that. Let me think. If I have √2 + 1/√2, I can write √2 as (√2 * √2)/√2, which is 2/√2. Wait, that's a good idea. Because √2 multiplied by √2 is 2, so √2 is equal to 2/√2. Therefore, √2 + 1/√2 is equal to 2/√2 + 1/√2, which is (2 + 1)/√2 = 3/√2. But then, I need to rationalize the denominator here. So 3/√2 is equal to 3√2 / 2. Okay, so that simplifies the √2 terms to 3√2 / 2.

Similarly, let me do the same for the √3 terms: √3 + 1/√3. Using the same method, √3 can be written as (√3 * √3)/√3 = 3/√3. Therefore, √3 + 1/√3 is 3/√3 + 1/√3 = 4/√3. Then, rationalizing the denominator, that becomes 4√3 / 3.

So now, putting it all together, the original expression is equal to 3√2 / 2 + 4√3 / 3. Now, I need to combine these two terms into a single fraction. To do that, I need a common denominator. The denominators here are 2 and 3, so the least common denominator is 6. Let me convert each term to have denominator 6.

Starting with 3√2 / 2: Multiply numerator and denominator by 3 to get (3√2 * 3) / (2 * 3) = 9√2 / 6.

Then, 4√3 / 3: Multiply numerator and denominator by 2 to get (4√3 * 2) / (3 * 2) = 8√3 / 6.

So now, adding these two fractions together: 9√2 / 6 + 8√3 / 6 = (9√2 + 8√3) / 6.

Therefore, the expression simplifies to (9√2 + 8√3)/6. Now, the problem states that we need to write it in the form (a√2 + b√3)/c where a, b, c are positive integers and c is as small as possible. Let me check if this fraction can be simplified further. The numerator is 9√2 + 8√3, and the denominator is 6.

To check if we can reduce the fraction, we need to see if there is a common factor in the coefficients of the numerator and the denominator. The coefficients are 9 and 8 in the numerator, and the denominator is 6. Let me check the greatest common divisor (GCD) of 9, 8, and 6.

The GCD of 9 and 8 is 1, since they are consecutive integers. Therefore, the GCD of 9, 8, and 6 is also 1. Therefore, there is no common factor to reduce the fraction further. Therefore, the simplified form is indeed (9√2 + 8√3)/6, so a = 9, b = 8, c = 6. Therefore, a + b + c = 9 + 8 + 6 = 23.

Wait, but let me double-check my steps to make sure I didn't make a mistake. Let me go through each step again.

Original expression: √2 + 1/√2 + √3 + 1/√3.

First, combining √2 and 1/√2:

√2 + 1/√2. As I did before, √2 is equal to 2/√2. Therefore, adding 2/√2 + 1/√2 = 3/√2. Rationalizing gives 3√2 / 2. That seems correct.

Similarly, √3 + 1/√3. √3 is 3/√3, so adding 3/√3 + 1/√3 = 4/√3. Rationalizing gives 4√3 / 3. That also seems correct.

Then adding 3√2 / 2 + 4√3 / 3. Common denominator is 6. Converting:

3√2 / 2 = (3√2 * 3)/6 = 9√2 / 6. Wait, hold on, 3 times 3 is 9? Wait, no. Wait, if I have 3√2 / 2, to get denominator 6, I multiply numerator and denominator by 3. So numerator becomes 3√2 * 3 = 9√2, denominator becomes 2 * 3 = 6. Yes, that's correct. Similarly, 4√3 / 3 multiplied by 2/2 gives 8√3 / 6. Then adding them gives (9√2 + 8√3)/6. That seems correct.

So then, the question is whether this fraction can be simplified further. The denominator is 6. The numerator is 9√2 + 8√3. Since √2 and √3 are irrational and linearly independent over the rationals, there's no way to combine them further. Therefore, the coefficients 9 and 8 in the numerator and denominator 6 have no common factors. Let me check the GCD of 9 and 6. GCD(9,6) is 3. But since the numerator has 9 and 8, and the denominator is 6, if we tried to factor out a 3 from the numerator and denominator, but the numerator has 9 and 8. If we factor out 3 from the 9, but 8 isn't divisible by 3. Therefore, we can't factor out a 3 from the entire numerator. Therefore, the fraction cannot be simplified further. Therefore, c is 6, which is as small as possible. Therefore, a=9, b=8, c=6, so a+b+c=23.

But let me check again if there's another way to combine the original terms that might lead to a different expression with a smaller denominator. Maybe I made a mistake in combining the terms?

Alternatively, perhaps I should have combined all terms together first? Let me try that approach to verify.

Original expression: √2 + 1/√2 + √3 + 1/√3. Let me write all terms with rational denominators first.

√2 is already fine, but 1/√2 is  √2/2. Similarly, √3 is fine, and 1/√3 is √3/3. Therefore, the entire expression can be written as:

√2 + √2/2 + √3 + √3/3.

Now, combining like terms:

For √2 terms: √2 + √2/2. Let me compute that. √2 is 2√2/2, so adding 2√2/2 + √2/2 = 3√2/2. Similarly, for √3 terms: √3 + √3/3. √3 is 3√3/3, so adding 3√3/3 + √3/3 = 4√3/3. Therefore, same as before. So adding those gives 3√2/2 + 4√3/3, which is the same as before. Therefore, same result. So that seems consistent.

Therefore, combining them gives (9√2 + 8√3)/6. Therefore, same conclusion. Therefore, a=9, b=8, c=6. Therefore, a+b+c=23.

But let me check if there's a different way to combine terms. For example, maybe combining √2 + √3 first? But that would not help, since they are different radicals. Similarly, combining 1/√2 + 1/√3? Let me try that.

Original expression: √2 + 1/√2 + √3 + 1/√3. Let me group √2 + √3 and 1/√2 + 1/√3. But that might not help. Let me see:

√2 + √3 + 1/√2 + 1/√3. If I try to combine 1/√2 + 1/√3, that would be (√3 + √2)/√6. Wait, let me check:

1/√2 + 1/√3 = (√3 + √2)/√6. Wait, let me verify:

Multiply numerator and denominator appropriately. Let me get common denominator √2√3 = √6. Therefore:

1/√2 = √3 / √6, and 1/√3 = √2 / √6. Therefore, adding them gives (√3 + √2)/√6. Therefore, that's another way to write the sum of the reciprocals. But then, the entire expression would be √2 + √3 + (√2 + √3)/√6. Hmm, not sure if that helps. Let me see:

So, if I write the entire expression as (√2 + √3) + (√2 + √3)/√6. Let me factor out (√2 + √3):

(√2 + √3)(1 + 1/√6). But that seems more complicated. Maybe not helpful. Let me try to compute this:

Let me compute (√2 + √3)(1 + 1/√6). Let me compute 1 + 1/√6. That is (√6 + 1)/√6. Therefore, multiplying by (√2 + √3):

(√2 + √3)(√6 + 1)/√6. Not sure if that helps. Maybe expanding the numerator?

Alternatively, perhaps this approach is more complicated. Let me check if this leads to a different expression. Let me compute:

First, compute (√2 + √3) + (√2 + √3)/√6. Let me write that as:

(√2 + √3) * [1 + 1/√6]. Let me compute 1 + 1/√6. Let me rationalize 1/√6 as √6/6. Therefore, 1 + √6/6 = (6 + √6)/6. Therefore, the entire expression becomes:

(√2 + √3) * (6 + √6)/6. Let me expand this:

Multiply (√2 + √3)(6 + √6). Let me compute that:

First, √2 * 6 = 6√2.

√2 * √6 = √(2*6) = √12 = 2√3.

√3 * 6 = 6√3.

√3 * √6 = √(3*6) = √18 = 3√2.

Therefore, adding all these terms:

6√2 + 2√3 + 6√3 + 3√2 = (6√2 + 3√2) + (2√3 + 6√3) = 9√2 + 8√3. Therefore, the entire expression is (9√2 + 8√3)/6, which is the same as before. Therefore, same result. Therefore, regardless of the method, I end up with the same expression. Therefore, that seems to confirm that the simplified form is indeed (9√2 + 8√3)/6. Therefore, a=9, b=8, c=6, so a+b+c=23.

But just to be thorough, let me check if there's a way to have a smaller denominator. Suppose someone thought that maybe combining terms differently could lead to a smaller denominator. Let me think. For example, if there was a common factor in numerator and denominator that I didn't consider. But since the numerator is 9√2 + 8√3, and denominator is 6. Let me check if 6 can be divided by some number that divides into both 9 and 8. But 9 and 8 have a GCD of 1, so there is no common factor. Therefore, the fraction cannot be simplified further. Therefore, c=6 is indeed the smallest possible denominator. Therefore, the answer is 23.

Alternatively, let me check if there's a different way to write the original expression. For example, maybe combining all terms over a common denominator from the start? Let me try that approach.

Original expression: √2 + 1/√2 + √3 + 1/√3. Let me write all terms with denominator √2√3, which is √6. Wait, but that might be complicated. Let me try:

First, write each term with denominator √6:

√2 can be written as √2 * √6 / √6 = √12 / √6 = 2√3 / √6. Wait, that seems messy. Alternatively, maybe not the best approach. Let me think.

Alternatively, write all terms over a common denominator. Let me think. The denominators are 1, √2, 1, √3. So the common denominator would be √2 * √3 = √6. Let me try that.

So, converting each term:

√2 = √2 * (√6 / √6) = (√2 * √6) / √6 = √12 / √6 = 2√3 / √6.

Similarly, 1/√2 = (√3 / √6).

√3 = √3 * (√6 / √6) = √18 / √6 = 3√2 / √6.

1/√3 = √2 / √6.

Therefore, adding all terms:

2√3 / √6 + √3 / √6 + 3√2 / √6 + √2 / √6.

Combine like terms:

For √3 terms: 2√3 + √3 = 3√3. So 3√3 / √6.

For √2 terms: 3√2 + √2 = 4√2. So 4√2 / √6.

Therefore, total expression is (3√3 + 4√2)/√6. Then, rationalizing the denominator:

Multiply numerator and denominator by √6:

[(3√3 + 4√2) * √6] / (√6 * √6) = [(3√3 * √6 + 4√2 * √6)] / 6.

Compute numerator:

First term: 3√3 * √6 = 3√(18) = 3*3√2 = 9√2.

Second term: 4√2 * √6 = 4√12 = 4*2√3 = 8√3.

Therefore, numerator is 9√2 + 8√3, denominator is 6. Therefore, same result as before: (9√2 + 8√3)/6. Therefore, same conclusion. Therefore, regardless of the method, the simplified form is the same. Therefore, I can be confident that the answer is 23.

But just to make sure, let me check if there's a different combination where maybe the denominator is smaller. For example, suppose someone thought that maybe combining terms differently could lead to cancellation. Let me think. For example, if I had written the original expression as:

√2 + √3 + 1/√2 + 1/√3. Suppose I tried to combine √2 + 1/√2 and √3 + 1/√3 as before, but maybe there's another way? Wait, I already did that. Alternatively, maybe combining √2 + √3 first? But that would be √2 + √3, which is irrational, and adding 1/√2 + 1/√3, which is also irrational. Not helpful.

Alternatively, maybe trying to write the entire expression as a single fraction. Let me try that. Let me consider all terms:

Original expression: √2 + 1/√2 + √3 + 1/√3. Let me write all terms over a common denominator. The denominators are 1, √2, 1, √3. So the common denominator would be √2 * √3 = √6. Wait, but that's what I did earlier. Which led me to the same result. Therefore, same conclusion.

Alternatively, maybe trying to combine terms in another way. For example, suppose I consider the entire expression as:

(√2 + 1/√2) + (√3 + 1/√3). Which is exactly what I did first. Which gives me 3√2 / 2 + 4√3 / 3. Then, combining over 6. So same result. Therefore, no alternative path here.

Alternatively, maybe there's a mistake in my calculation of combining the terms? Let me check again:

Original terms:

√2 + 1/√2. Let me compute decimal approximations to check if my algebra is correct. Let me compute √2 ≈ 1.4142, 1/√2 ≈ 0.7071. Adding them together: 1.4142 + 0.7071 ≈ 2.1213. Now, 3√2 / 2 ≈ 3 * 1.4142 / 2 ≈ 4.2426 / 2 ≈ 2.1213. Correct. Similarly, √3 ≈ 1.732, 1/√3 ≈ 0.577. Adding them gives ≈ 2.309. 4√3 / 3 ≈ 4 * 1.732 / 3 ≈ 6.928 / 3 ≈ 2.309. Correct. Then adding 2.1213 + 2.309 ≈ 4.4303. Now, let me compute (9√2 + 8√3)/6. Let me compute numerator:

9√2 ≈ 9 * 1.4142 ≈ 12.7278

8√3 ≈ 8 * 1.732 ≈ 13.856

Adding them: 12.7278 + 13.856 ≈ 26.5838

Divide by 6: ≈ 26.5838 / 6 ≈ 4.4306. Which matches the approximate value of the original expression. Therefore, that seems correct. Therefore, my algebra is correct.

Therefore, I think my answer is correct. Therefore, the answer is 23.

But just to be 100% sure, let me check if there's a way to have a smaller denominator. Suppose someone thought that maybe 9 and 8 have a common factor with 6? For example, if I can factor something out. Let me think. The denominator is 6. The numerator is 9√2 + 8√3. If I factor out a 1, there's nothing. If I factor out a 3 from the 9√2 term, I get 3*(3√2) + 8√3. But then denominator is 6, so if I factor out 3 from numerator and denominator, I would have [3*(3√2) + 8√3]/6 = [3*(3√2) + 8√3]/6. But since 8√3 isn't divisible by 3, I can't factor out a 3 from the entire numerator. Therefore, that doesn't help. Similarly, trying to factor out 2? 9√2 is divisible by 2? No, 9 is odd. 8√3 is divisible by 2. So if I factor out 2 from the 8√3 term, I get 9√2 + 2*(4√3). Then denominator is 6. So if I factor out 2 from numerator and denominator, I would have [9√2 + 2*(4√3)] / 6 = [9√2 + 8√3]/6. Which is the same as before. Therefore, no help. Therefore, there's no way to factor out anything from numerator and denominator. Therefore, the fraction is in simplest terms. Therefore, c=6 is indeed minimal.

Therefore, the answer is 9 + 8 + 6 = 23. Therefore, I think that's the correct answer.

**Final Answer**
\boxed{23}

To simplify the expression
$$
\sqrt{2} + \frac{1}{\sqrt{2}} + \sqrt{3} + \frac{1}{\sqrt{3}}
$$
into the form
$$
\frac{a\sqrt{2} + b\sqrt{3}}{c}
$$
with $ a, b, c $ as positive integers and $ c $ as small as possible, we proceed step by step.

---

### Step 1: Simplify each pair of terms

We group the terms with the same radical:

- **For $ \sqrt{2} $:**
  $$
  \sqrt{2} + \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{3}{\sqrt{2}} = \frac{3\sqrt{2}}{2}
  $$

- **For $ \sqrt{3} $:**
  $$
  \sqrt{3} + \frac{1}{\sqrt{3}} = \frac{3}{\sqrt{3}} + \frac{1}{\sqrt{3}} = \frac{4}{\sqrt{3}} = \frac{4\sqrt{3}}{3}
  $$

---

### Step 2: Combine the simplified terms

Now we have:
$$
\frac{3\sqrt{2}}{2} + \frac{4\sqrt{3}}{3}
$$

To add these, we find a common denominator. The least common denominator of 2 and 3 is 6.

- Convert $ \frac{3\sqrt{2}}{2} $ to denominator 6:
  $$
  \frac{3\sqrt{2}}{2} = \frac{9\sqrt{2}}{6}
  $$

- Convert $ \frac{4\sqrt{3}}{3} $ to denominator 6:
  $$
  \frac{4\sqrt{3}}{3} = \frac{8\sqrt{3}}{6}
  $$

Now add the two:
$$
\frac{9\sqrt{2}}{6} + \frac{8\sqrt{3}}{6} = \frac{9\sqrt{2} + 8\sqrt{3}}{6}
$$

---

### Step 3: Check for simplification

The numerator is $ 9\sqrt{2} + 8\sqrt{3} $, and the denominator is 6. Since $ \sqrt{2} $ and $ \sqrt{3} $ are linearly independent over the rationals, the expression cannot be simplified further. The greatest common divisor of the coefficients 9, 8, and 6 is 1, so the fraction is already in its simplest form.

---

### Final Answer

We now have:
$$
\frac{9\sqrt{2} + 8\sqrt{3}}{6}
$$

So, $ a = 9 $, $ b = 8 $, and $ c = 6 $. Therefore:

$$
a + b + c = 9 + 8 + 6 = \boxed{23}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how to write the expression √2 + 1/√2 + √3 + 1/√3 in the form (a√2 + b√3)/c, where a, b, and c are positive integers, and c is as small as possible. Then, I have to find a + b + c. Alright, let me start by breaking down the problem step by step.

First, let me look at each term individually. The expression has four terms: √2, 1/√2, √3, and 1/√3. Maybe I can combine the terms with √2 and the terms with √3 separately. That might make it easier to handle.

Starting with the √2 terms: √2 and 1/√2. Let me combine these two. To add them together, they need a common denominator. The term √2 can be written as √2/1, and 1/√2 is already a fraction. To add them, I can rationalize the denominator of 1/√2. Wait, but maybe I can just combine them directly?

Alternatively, maybe I can combine them by converting them to have the same denominator. Let me try that. Let me think. If I have √2 + 1/√2, I can write √2 as (√2 * √2)/√2, which is 2/√2. Wait, that's a good idea. Because √2 multiplied by √2 is 2, so √2 is equal to 2/√2. Therefore, √2 + 1/√2 is equal to 2/√2 + 1/√2, which is (2 + 1)/√2 = 3/√2. But then, I need to rationalize the denominator here. So 3/√2 is equal to 3√2 / 2. Okay, so that simplifies the √2 terms to 3√2 / 2.

Similarly, let me do the same for the √3 terms: √3 + 1/√3. Using the same method, √3 can be written as (√3 * √3)/√3 = 3/√3. Therefore, √3 + 1/√3 is 3/√3 + 1/√3 = 4/√3. Then, rationalizing the denominator, that becomes 4√3 / 3.

So now, putting it all together, the original expression is equal to 3√2 / 2 + 4√3 / 3. Now, I need to combine these two terms into a single fraction. To do that, I need a common denominator. The denominators here are 2 and 3, so the least common denominator is 6. Let me convert each term to have denominator 6.

Starting with 3√2 / 2: Multiply numerator and denominator by 3 to get (3√2 * 3) / (2 * 3) = 9√2 / 6.

Then, 4√3 / 3: Multiply numerator and denominator by 2 to get (4√3 * 2) / (3 * 2) = 8√3 / 6.

So now, adding these two fractions together: 9√2 / 6 + 8√3 / 6 = (9√2 + 8√3) / 6.

Therefore, the expression simplifies to (9√2 + 8√3)/6. Now, the problem states that we need to write it in the form (a√2 + b√3)/c where a, b, c are positive integers and c is as small as possible. Let me check if this fraction can be simplified further. The numerator is 9√2 + 8√3, and the denominator is 6.

To check if we can reduce the fraction, we need to see if there is a common factor in the coefficients of the numerator and the denominator. The coefficients are 9 and 8 in the numerator, and the denominator is 6. Let me check the greatest common divisor (GCD) of 9, 8, and 6.

The GCD of 9 and 8 is 1, since they are consecutive integers. Therefore, the GCD of 9, 8, and 6 is also 1. Therefore, there is no common factor to reduce the fraction further. Therefore, the simplified form is indeed (9√2 + 8√3)/6, so a = 9, b = 8, c = 6. Therefore, a + b + c = 9 + 8 + 6 = 23.

Wait, but let me double-check my steps to make sure I didn't make a mistake. Let me go through each step again.

Original expression: √2 + 1/√2 + √3 + 1/√3.

First, combining √2 and 1/√2:

√2 + 1/√2. As I did before, √2 is equal to 2/√2. Therefore, adding 2/√2 + 1/√2 = 3/√2. Rationalizing gives 3√2 / 2. That seems correct.

Similarly, √3 + 1/√3. √3 is 3/√3, so adding 3/√3 + 1/√3 = 4/√3. Rationalizing gives 4√3 / 3. That also seems correct.

Then adding 3√2 / 2 + 4√3 / 3. Common denominator is 6. Converting:

3√2 / 2 = (3√2 * 3)/6 = 9√2 / 6. Wait, hold on, 3 times 3 is 9? Wait, no. Wait, if I have 3√2 / 2, to get denominator 6, I multiply numerator and denominator by 3. So numerator becomes 3√2 * 3 = 9√2, denominator becomes 2 * 3 = 6. Yes, that's correct. Similarly, 4√3 / 3 multiplied by 2/2 gives 8√3 / 6. Then adding them gives (9√2 + 8√3)/6. That seems correct.

So then, the question is whether this fraction can be simplified further. The denominator is 6. The numerator is 9√2 + 8√3. Since √2 and √3 are irrational and linearly independent over the rationals, there's no way to combine them further. Therefore, the coefficients 9 and 8 in the numerator and denominator 6 have no common factors. Let me check the GCD of 9 and 6. GCD(9,6) is 3. But since the numerator has 9 and 8, and the denominator is 6, if we tried to factor out a 3 from the numerator and denominator, but the numerator has 9 and 8. If we factor out 3 from the 9, but 8 isn't divisible by 3. Therefore, we can't factor out a 3 from the entire numerator. Therefore, the fraction cannot be simplified further. Therefore, c is 6, which is as small as possible. Therefore, a=9, b=8, c=6, so a+b+c=23.

But let me check again if there's another way to combine the original terms that might lead to a different expression with a smaller denominator. Maybe I made a mistake in combining the terms?

Alternatively, perhaps I should have combined all terms together first? Let me try that approach to verify.

Original expression: √2 + 1/√2 + √3 + 1/√3. Let me write all terms with rational denominators first.

√2 is already fine, but 1/√2 is  √2/2. Similarly, √3 is fine, and 1/√3 is √3/3. Therefore, the entire expression can be written as:

√2 + √2/2 + √3 + √3/3.

Now, combining like terms:

For √2 terms: √2 + √2/2. Let me compute that. √2 is 2√2/2, so adding 2√2/2 + √2/2 = 3√2/2. Similarly, for √3 terms: √3 + √3/3. √3 is 3√3/3, so adding 3√3/3 + √3/3 = 4√3/3. Therefore, same as before. So adding those gives 3√2/2 + 4√3/3, which is the same as before. Therefore, same result. So that seems consistent.

Therefore, combining them gives (9√2 + 8√3)/6. Therefore, same conclusion. Therefore, a=9, b=8, c=6. Therefore, a+b+c=23.

But let me check if there's a different way to combine terms. For example, maybe combining √2 + √3 first? But that would not help, since they are different radicals. Similarly, combining 1/√2 + 1/√3? Let me try that.

Original expression: √2 + 1/√2 + √3 + 1/√3. Let me group √2 + √3 and 1/√2 + 1/√3. But that might not help. Let me see:

√2 + √3 + 1/√2 + 1/√3. If I try to combine 1/√2 + 1/√3, that would be (√3 + √2)/√6. Wait, let me check:

1/√2 + 1/√3 = (√3 + √2)/√6. Wait, let me verify:

Multiply numerator and denominator appropriately. Let me get common denominator √2√3 = √6. Therefore:

1/√2 = √3 / √6, and 1/√3 = √2 / √6. Therefore, adding them gives (√3 + √2)/√6. Therefore, that's another way to write the sum of the reciprocals. But then, the entire expression would be √2 + √3 + (√2 + √3)/√6. Hmm, not sure if that helps. Let me see:

So, if I write the entire expression as (√2 + √3) + (√2 + √3)/√6. Let me factor out (√2 + √3):

(√2 + √3)(1 + 1/√6). But that seems more complicated. Maybe not helpful. Let me try to compute this:

Let me compute (√2 + √3)(1 + 1/√6). Let me compute 1 + 1/√6. That is (√6 + 1)/√6. Therefore, multiplying by (√2 + √3):

(√2 + √3)(√6 + 1)/√6. Not sure if that helps. Maybe expanding the numerator?

Alternatively, perhaps this approach is more complicated. Let me check if this leads to a different expression. Let me compute:

First, compute (√2 + √3) + (√2 + √3)/√6. Let me write that as:

(√2 + √3) * [1 + 1/√6]. Let me compute 1 + 1/√6. Let me rationalize 1/√6 as √6/6. Therefore, 1 + √6/6 = (6 + √6)/6. Therefore, the entire expression becomes:

(√2 + √3) * (6 + √6)/6. Let me expand this:

Multiply (√2 + √3)(6 + √6). Let me compute that:

First, √2 * 6 = 6√2.

√2 * √6 = √(2*6) = √12 = 2√3.

√3 * 6 = 6√3.

√3 * √6 = √(3*6) = √18 = 3√2.

Therefore, adding all these terms:

6√2 + 2√3 + 6√3 + 3√2 = (6√2 + 3√2) + (2√3 + 6√3) = 9√2 + 8√3. Therefore, the entire expression is (9√2 + 8√3)/6, which is the same as before. Therefore, same result. Therefore, regardless of the method, I end up with the same expression. Therefore, that seems to confirm that the simplified form is indeed (9√2 + 8√3)/6. Therefore, a=9, b=8, c=6, so a+b+c=23.

But just to be thorough, let me check if there's a way to have a smaller denominator. Suppose someone thought that maybe combining terms differently could lead to a smaller denominator. Let me think. For example, if there was a common factor in numerator and denominator that I didn't consider. But since the numerator is 9√2 + 8√3, and denominator is 6. Let me check if 6 can be divided by some number that divides into both 9 and 8. But 9 and 8 have a GCD of 1, so there is no common factor. Therefore, the fraction cannot be simplified further. Therefore, c=6 is indeed the smallest possible denominator. Therefore, the answer is 23.

Alternatively, let me check if there's a different way to write the original expression. For example, maybe combining all terms over a common denominator from the start? Let me try that approach.

Original expression: √2 + 1/√2 + √3 + 1/√3. Let me write all terms with denominator √2√3, which is √6. Wait, but that might be complicated. Let me try:

First, write each term with denominator √6:

√2 can be written as √2 * √6 / √6 = √12 / √6 = 2√3 / √6. Wait, that seems messy. Alternatively, maybe not the best approach. Let me think.

Alternatively, write all terms over a common denominator. Let me think. The denominators are 1, √2, 1, √3. So the common denominator would be √2 * √3 = √6. Let me try that.

So, converting each term:

√2 = √2 * (√6 / √6) = (√2 * √6) / √6 = √12 / √6 = 2√3 / √6.

Similarly, 1/√2 = (√3 / √6).

√3 = √3 * (√6 / √6) = √18 / √6 = 3√2 / √6.

1/√3 = √2 / √6.

Therefore, adding all terms:

2√3 / √6 + √3 / √6 + 3√2 / √6 + √2 / √6.

Combine like terms:

For √3 terms: 2√3 + √3 = 3√3. So 3√3 / √6.

For √2 terms: 3√2 + √2 = 4√2. So 4√2 / √6.

Therefore, total expression is (3√3 + 4√2)/√6. Then, rationalizing the denominator:

Multiply numerator and denominator by √6:

[(3√3 + 4√2) * √6] / (√6 * √6) = [(3√3 * √6 + 4√2 * √6)] / 6.

Compute numerator:

First term: 3√3 * √6 = 3√(18) = 3*3√2 = 9√2.

Second term: 4√2 * √6 = 4√12 = 4*2√3 = 8√3.

Therefore, numerator is 9√2 + 8√3, denominator is 6. Therefore, same result as before: (9√2 + 8√3)/6. Therefore, same conclusion. Therefore, regardless of the method, the simplified form is the same. Therefore, I can be confident that the answer is 23.

But just to make sure, let me check if there's a different combination where maybe the denominator is smaller. For example, suppose someone thought that maybe combining terms differently could lead to cancellation. Let me think. For example, if I had written the original expression as:

√2 + √3 + 1/√2 + 1/√3. Suppose I tried to combine √2 + 1/√2 and √3 + 1/√3 as before, but maybe there's another way? Wait, I already did that. Alternatively, maybe combining √2 + √3 first? But that would be √2 + √3, which is irrational, and adding 1/√2 + 1/√3, which is also irrational. Not helpful.

Alternatively, maybe trying to write the entire expression as a single fraction. Let me try that. Let me consider all terms:

Original expression: √2 + 1/√2 + √3 + 1/√3. Let me write all terms over a common denominator. The denominators are 1, √2, 1, √3. So the common denominator would be √2 * √3 = √6. Wait, but that's what I did earlier. Which led me to the same result. Therefore, same conclusion.

Alternatively, maybe trying to combine terms in another way. For example, suppose I consider the entire expression as:

(√2 + 1/√2) + (√3 + 1/√3). Which is exactly what I did first. Which gives me 3√2 / 2 + 4√3 / 3. Then, combining over 6. So same result. Therefore, no alternative path here.

Alternatively, maybe there's a mistake in my calculation of combining the terms? Let me check again:

Original terms:

√2 + 1/√2. Let me compute decimal approximations to check if my algebra is correct. Let me compute √2 ≈ 1.4142, 1/√2 ≈ 0.7071. Adding them together: 1.4142 + 0.7071 ≈ 2.1213. Now, 3√2 / 2 ≈ 3 * 1.4142 / 2 ≈ 4.2426 / 2 ≈ 2.1213. Correct. Similarly, √3 ≈ 1.732, 1/√3 ≈ 0.577. Adding them gives ≈ 2.309. 4√3 / 3 ≈ 4 * 1.732 / 3 ≈ 6.928 / 3 ≈ 2.309. Correct. Then adding 2.1213 + 2.309 ≈ 4.4303. Now, let me compute (9√2 + 8√3)/6. Let me compute numerator:

9√2 ≈ 9 * 1.4142 ≈ 12.7278

8√3 ≈ 8 * 1.732 ≈ 13.856

Adding them: 12.7278 + 13.856 ≈ 26.5838

Divide by 6: ≈ 26.5838 / 6 ≈ 4.4306. Which matches the approximate value of the original expression. Therefore, that seems correct. Therefore, my algebra is correct.

Therefore, I think my answer is correct. Therefore, the answer is 23.

But just to be 100% sure, let me check if there's a way to have a smaller denominator. Suppose someone thought that maybe 9 and 8 have a common factor with 6? For example, if I can factor something out. Let me think. The denominator is 6. The numerator is 9√2 + 8√3. If I factor out a 1, there's nothing. If I factor out a 3 from the 9√2 term, I get 3*(3√2) + 8√3. But then denominator is 6, so if I factor out 3 from numerator and denominator, I would have [3*(3√2) + 8√3]/6 = [3*(3√2) + 8√3]/6. But since 8√3 isn't divisible by 3, I can't factor out a 3 from the entire numerator. Therefore, that doesn't help. Similarly, trying to factor out 2? 9√2 is divisible by 2? No, 9 is odd. 8√3 is divisible by 2. So if I factor out 2 from the 8√3 term, I get 9√2 + 2*(4√3). Then denominator is 6. So if I factor out 2 from numerator and denominator, I would have [9√2 + 2*(4√3)] / 6 = [9√2 + 8√3]/6. Which is the same as before. Therefore, no help. Therefore, there's no way to factor out anything from numerator and denominator. Therefore, the fraction is in simplest terms. Therefore, c=6 is indeed minimal.

Therefore, the answer is 9 + 8 + 6 = 23. Therefore, I think that's the correct answer.

**Final Answer**
\boxed{23}

To simplify the expression
$$
\sqrt{2} + \frac{1}{\sqrt{2}} + \sqrt{3} + \frac{1}{\sqrt{3}}
$$
into the form
$$
\frac{a\sqrt{2} + b\sqrt{3}}{c}
$$
with $ a, b, c $ as positive integers and $ c $ as small as possible, we proceed step by step.

---

### Step 1: Simplify each pair of terms

We group the terms with the same radical:

- **For $ \sqrt{2} $:**
  $$
  \sqrt{2} + \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \frac{3}{\sqrt{2}} = \frac{3\sqrt{2}}{2}
  $$

- **For $ \sqrt{3} $:**
  $$
  \sqrt{3} + \frac{1}{\sqrt{3}} = \frac{3}{\sqrt{3}} + \frac{1}{\sqrt{3}} = \frac{4}{\sqrt{3}} = \frac{4\sqrt{3}}{3}
  $$

---

### Step 2: Combine the simplified terms

Now we have:
$$
\frac{3\sqrt{2}}{2} + \frac{4\sqrt{3}}{3}
$$

To add these, we find a common denominator. The least common denominator of 2 and 3 is 6.

- Convert $ \frac{3\sqrt{2}}{2} $ to denominator 6:
  $$
  \frac{3\sqrt{2}}{2} = \frac{9\sqrt{2}}{6}
  $$

- Convert $ \frac{4\sqrt{3}}{3} $ to denominator 6:
  $$
  \frac{4\sqrt{3}}{3} = \frac{8\sqrt{3}}{6}
  $$

Now add the two:
$$
\frac{9\sqrt{2}}{6} + \frac{8\sqrt{3}}{6} = \frac{9\sqrt{2} + 8\sqrt{3}}{6}
$$

---

### Step 3: Check for simplification

The numerator is $ 9\sqrt{2} + 8\sqrt{3} $, and the denominator is 6. Since $ \sqrt{2} $ and $ \sqrt{3} $ are linearly independent over the rationals, the expression cannot be simplified further. The greatest common divisor of the coefficients 9, 8, and 6 is 1, so the fraction is already in its simplest form.

---

### Final Answer

We now have:
$$
\frac{9\sqrt{2} + 8\sqrt{3}}{6}
$$

So, $ a = 9 $, $ b = 8 $, and $ c = 6 $. Therefore:

$$
a + b + c = 9 + 8 + 6 = \boxed{23}
$$

Solution: The common denominator desired is $\sqrt{2}\cdot\sqrt{3} = \sqrt{6}$. So, this expression becomes $\frac{\sqrt{2}\cdot(\sqrt{2}\cdot\sqrt{3})+1\cdot\sqrt{3}+\sqrt{3}\cdot(\sqrt{2}\cdot\sqrt{3})+1\cdot\sqrt{2}}{\sqrt{6}}$. Simplifying this gives $\frac{2\sqrt{3}+\sqrt{3}+3\sqrt{2}+\sqrt{2}}{\sqrt{6}} = \frac{4\sqrt{2}+3\sqrt{3}}{\sqrt{6}}$. To rationalize, multiply numerator and denominator by $\sqrt{6}$ to get $\frac{4\sqrt{2}\sqrt{6}+3\sqrt{3}\sqrt{6}}{6}$. Simplifying yields ${\frac{9\sqrt{2}+8\sqrt{3}}{6}}$, so the desired sum is $9+8+6=\boxed{23}$.


Explanation:
Okay, so I need to figure out for how many positive integers n > 1 it's true that 2^24 is a perfect n-th power. Hmm, let me start by recalling what a perfect n-th power means. If a number is a perfect n-th power, that means it can be written as some integer raised to the n-th power. So, in this case, 2^24 should be equal to some integer k raised to the n-th power, right?

So, mathematically, I can write that as 2^24 = k^n. Now, since 2 is a prime number, maybe I can use prime factorization here. The prime factorization of 2^24 is just 2^24, so that's straightforward.

Now, if 2^24 is equal to k^n, then k must be a power of 2 as well, because the prime factors on the right side have to match those on the left. Let me think. If k had any other prime factors, then the prime factorization of k^n would include those primes, which aren't in 2^24. Therefore, k must be of the form 2^m, where m is some positive integer.

So substituting back into the equation, we get 2^24 = (2^m)^n = 2^{m*n}. Therefore, the exponents must be equal. That gives us the equation m*n = 24.

So, m and n are positive integers such that m*n = 24. But we need to find the number of positive integers n > 1 for which this is possible. Wait, but m is also a positive integer. So, for each n > 1 that divides 24, there exists an m such that m = 24/n. Therefore, the number of such n is equal to the number of divisors of 24 that are greater than 1.

But hold on, let me make sure. Since m has to be a positive integer, n must be a divisor of 24. So, the problem reduces to finding the number of divisors of 24 that are greater than 1.

But let me check if there are any restrictions on m. Since k is a positive integer, m must be a positive integer as well. Therefore, n must be a positive integer divisor of 24. However, the problem states n > 1, so we need to exclude n = 1.

So, first, let me find all the divisors of 24. Let me list them out. The prime factorization of 24 is 2^3 * 3^1. The number of divisors is (3 + 1)*(1 + 1) = 4*2 = 8. So there are 8 divisors. Let me list them:

1, 2, 3, 4, 6, 8, 12, 24.

So, the divisors greater than 1 are 2, 3, 4, 6, 8, 12, 24. That's 7 divisors. Therefore, there are 7 positive integers n > 1 for which 2^24 is a perfect n-th power.

Wait, but hold on. Let me think again. Is there a possibility that for some n, even though n divides 24, there might not be an integer k? But earlier, we saw that if n divides 24, then m = 24/n is an integer, so k = 2^m is an integer. Therefore, for each divisor n > 1 of 24, there exists such a k. Therefore, the answer should be 7.

But let me check with examples. Let's take n=2. Then 2^24 is (2^12)^2, which is 4096^2. That works. For n=3, 2^24 is (2^8)^3, since 8*3=24. 2^8 is 256, so 256^3. That works. For n=4, 2^24 is (2^6)^4, since 6*4=24. 2^6 is 64, so 64^4. That works. For n=6, 2^24 is (2^4)^6, since 4*6=24. 2^4 is 16, so 16^6. That works. For n=8, 2^24 is (2^3)^8, since 3*8=24. 2^3 is 8, so 8^8. That works. For n=12, 2^24 is (2^2)^12, since 2*12=24. 2^2 is 4, so 4^12. That works. For n=24, 2^24 is (2^1)^24, which is 2^24. So that's 2^24. So that works.

So all of these n's work. Therefore, there are 7 such n's.

But wait, let me check if there are any other n's that might work. For example, suppose n is not a divisor of 24. Let's say n=5. Then, can 2^24 be written as k^5? Let's see. If so, then k^5 = 2^24. Then, k would have to be 2^m, so m*5 = 24. But 24 divided by 5 is 4.8, which is not an integer. Therefore, there is no integer m such that m*5=24, so n=5 is not valid. Similarly, n=7: 24 divided by 7 is not integer. So, only divisors of 24 will work. Therefore, the answer is indeed 7.

But let me check if there's a different approach. Maybe considering exponents. Since 2^24 is a perfect n-th power, then the exponent 24 must be divisible by n. Therefore, n must divide 24. Therefore, the number of n>1 is equal to the number of divisors of 24 greater than 1, which is 7.

Alternatively, if I think about the exponents in the prime factorization. For a number to be a perfect n-th power, all exponents in its prime factorization must be multiples of n. In this case, the prime factorization of 2^24 is just 2^24. Therefore, the exponent 24 must be a multiple of n. Therefore, n divides 24. Therefore, same conclusion.

Therefore, the answer is 7.

But let me check once more. The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Excluding 1, we have 7 numbers. Therefore, the answer is 7.

But wait, the problem says "positive integers n > 1". So, n can be 2,3,4,6,8,12,24. That's 7 numbers. Therefore, the answer is 7.

But let me check if there's a possibility that for some n, even though n divides 24, the number might not be a perfect n-th power? For example, suppose n=2. Then 2^24 is (2^12)^2, which is a perfect square. Similarly, n=3, (2^8)^3. So, yes, all of them are perfect n-th powers.

Alternatively, if n is a divisor of 24, then 24/n is integer, so 2^{24/n} is integer, so (2^{24/n})^n = 2^{24}. Therefore, yes, that works.

Therefore, I think my reasoning is correct. Therefore, the answer is 7.

But wait, hold on. Let me check if n can be something else. For example, suppose n is a composite number. Wait, but n is already a divisor of 24. So, for example, n=4, which is composite. But even if n is composite, as long as it divides 24, it's okay.

Alternatively, maybe there's a case where n is not a divisor of 24, but still 2^24 is a perfect n-th power? For example, suppose n= something like 24/k where k is... Wait, no. If 2^24 is a perfect n-th power, then n must divide 24. Because if you have 2^24 = k^n, then k must be a power of 2, say 2^m, so 2^{24} = 2^{m*n}, so 24 = m*n. Therefore, n divides 24. Therefore, n must be a divisor of 24. Therefore, there are no other possibilities.

Therefore, the answer is 7.

But let me check once again. Let me list all the divisors of 24 greater than 1:

Divisors of 24: 1, 2, 3, 4, 6, 8, 12, 24. So removing 1, we have 2, 3, 4, 6, 8, 12, 24. That's 7 numbers. Therefore, the answer is 7.

Therefore, I think the answer is 7.

But wait, hold on. Wait, the problem says "for how many positive integers n > 1". So, n must be greater than 1. So, 7 numbers.

But let me check if there's a mistake in considering exponents. For example, suppose n= something like 24. Then, 2^24 is (2^1)^24, which is 2^24. So that's a perfect 24th power. Similarly, n=12, 2^24 is (2^2)^12. So, yes.

Alternatively, if n= something like 24/k, but no, n is already a divisor.

Alternatively, maybe there's a case where n is not a divisor of 24, but 2^24 is still a perfect n-th power? For example, suppose n= something like 24 divided by a fraction? But n has to be an integer.

Alternatively, suppose n is a divisor of 24, but maybe there's some other way? For example, if n is  something else. Wait, no. Because if 2^24 is a perfect n-th power, then n must divide 24.

Therefore, I think my answer is correct.

But let me check with another example. Let's take n= something like 24. If n=24, then 2^24 is 2^(24) which is (2^1)^24. So that's a perfect 24th power. Similarly, n=12, 2^24 is (2^2)^12. So, yes.

Alternatively, if n= something like 24/ something. Wait, but n is already a divisor.

Alternatively, maybe n is not a divisor, but 24 is a multiple of n. Wait, but that's the same thing. If n divides 24, then 24 is a multiple of n.

Therefore, I think my answer is correct. Therefore, the number of positive integers n>1 is 7.

But let me check once again. Let me count the divisors again.

Divisors of 24:

Start with 1. Then 2, 3, 4, 6, 8, 12, 24. So that's 8 divisors. Excluding 1, 7 divisors. Therefore, 7 numbers.

Therefore, the answer is 7.

But wait, hold on. Wait, the problem says "perfect n-th power". Is there a case where even if n divides 24, the number might not be a perfect n-th power? For example, suppose n= something. Wait, no. Because if n divides 24, then 24 = n * m, so 2^24 = (2^m)^n. Therefore, it is a perfect n-th power. Therefore, all divisors n>1 of 24 will satisfy the condition.

Therefore, the answer is 7.

But let me check if the problem is maybe considering exponents in a different way. For example, maybe considering that 2^24 is a perfect n-th power if and only if n divides 24. Therefore, the number of n>1 is equal to the number of divisors of 24 greater than 1, which is 7.

Alternatively, maybe there's a mistake in considering that k must be an integer. For example, if n divides 24, then m=24/n is integer, so k=2^m is integer. Therefore, yes.

Therefore, I think my answer is correct.

But just to be thorough, let me check for n=2,3,4,6,8,12,24:

n=2: 2^24 = (2^12)^2. Yes.

n=3: 2^24 = (2^8)^3. Yes.

n=4: 2^24 = (2^6)^4. Yes.

n=6: 2^24 = (2^4)^6. Yes.

n=8: 2^24 = (2^3)^8. Yes.

n=12: 2^24 = (2^2)^12. Yes.

n=24: 2^24 = (2^1)^24. Yes.

All of these work. Therefore, 7 values.

Therefore, the answer is 7.

But wait, hold on. Wait, the problem says "positive integers n>1". So, n=2,3,4,6,8,12,24. That's 7 numbers. Therefore, the answer is 7.

But let me check if there's a different interpretation. For example, maybe the question is asking for how many n>1 such that 2^24 is a perfect n-th power, but maybe considering different exponents? For example, maybe n is not necessarily a divisor of 24? But no, as we saw earlier, if 2^24 is a perfect n-th power, then n must divide 24.

Alternatively, maybe considering exponents in a different way. For example, if 2^24 is a perfect n-th power, then there exists some integer k such that k^n = 2^24. Then, since 2 is prime, k must be a power of 2, so k=2^m, so (2^m)^n = 2^{m*n} = 2^24. Therefore, m*n=24. Therefore, n must divide 24. Therefore, same conclusion.

Therefore, the answer is 7.

Therefore, I think I can be confident that the answer is 7.

**Final Answer**
\boxed{7}

To determine how many positive integers $ n > 1 $ make $ 2^{24} $ a perfect $ n $-th power, we begin by understanding what it means for a number to be a perfect $ n $-th power.

---

### Step 1: Understanding the Definition

A number $ x $ is a **perfect $ n $-th power** if there exists some integer $ k $ such that:

$$
x = k^n
$$

In this case, we are given $ x = 2^{24} $, and we want to find all integers $ n > 1 $ such that:

$$
2^{24} = k^n
$$

---

### Step 2: Prime Factorization Insight

Since $ 2^{24} $ is a power of a prime number, any $ k $ that satisfies $ k^n = 2^{24} $ must also be a power of 2. Let’s suppose:

$$
k = 2^m
$$

Then:

$$
k^n = (2^m)^n = 2^{mn}
$$

So we have:

$$
2^{mn} = 2^{24} \Rightarrow mn = 24
$$

This tells us that $ n $ must be a **positive divisor of 24**, and $ m = \frac{24}{n} $ must also be a positive integer.

---

### Step 3: Counting Valid $ n > 1 $

We now find all **positive divisors of 24** and exclude $ n = 1 $, since the problem specifies $ n > 1 $.

The prime factorization of 24 is:

$$
24 = 2^3 \cdot 3^1
$$

The number of positive divisors is:

$$
(3 + 1)(1 + 1) = 4 \cdot 2 = 8
$$

The divisors of 24 are:

$$
1, 2, 3, 4, 6, 8, 12, 24
$$

Excluding $ n = 1 $, the valid values of $ n $ are:

$$
2, 3, 4, 6, 8, 12, 24
$$

That’s a total of **7** values.

---

### Step 4: Verifying Each Case

For each of these $ n $, we can write:

$$
2^{24} = (2^{24/n})^n
$$

Since $ 24/n $ is an integer for each of these $ n $, the expression is valid, and $ 2^{24} $ is indeed a perfect $ n $-th power.

---

### Final Answer

$$
\boxed{7}
$$

[Assistant]
Explanation:
Okay, so I need to figure out for how many positive integers n > 1 it's true that 2^24 is a perfect n-th power. Hmm, let me start by recalling what a perfect n-th power means. If a number is a perfect n-th power, that means it can be written as some integer raised to the n-th power. So, in this case, 2^24 should be equal to some integer k raised to the n-th power, right?

So, mathematically, I can write that as 2^24 = k^n. Now, since 2 is a prime number, maybe I can use prime factorization here. The prime factorization of 2^24 is just 2^24, so that's straightforward.

Now, if 2^24 is equal to k^n, then k must be a power of 2 as well, because the prime factors on the right side have to match those on the left. Let me think. If k had any other prime factors, then the prime factorization of k^n would include those primes, which aren't in 2^24. Therefore, k must be of the form 2^m, where m is some positive integer.

So substituting back into the equation, we get 2^24 = (2^m)^n = 2^{m*n}. Therefore, the exponents must be equal. That gives us the equation m*n = 24.

So, m and n are positive integers such that m*n = 24. But we need to find the number of positive integers n > 1 for which this is possible. Wait, but m is also a positive integer. So, for each n > 1 that divides 24, there exists an m such that m = 24/n. Therefore, the number of such n is equal to the number of divisors of 24 that are greater than 1.

But hold on, let me make sure. Since m has to be a positive integer, n must be a divisor of 24. So, the problem reduces to finding the number of divisors of 24 that are greater than 1.

But let me check if there are any restrictions on m. Since k is a positive integer, m must be a positive integer as well. Therefore, n must be a positive integer divisor of 24. However, the problem states n > 1, so we need to exclude n = 1.

So, first, let me find all the divisors of 24. Let me list them out. The prime factorization of 24 is 2^3 * 3^1. The number of divisors is (3 + 1)*(1 + 1) = 4*2 = 8. So there are 8 divisors. Let me list them:

1, 2, 3, 4, 6, 8, 12, 24.

So, the divisors greater than 1 are 2, 3, 4, 6, 8, 12, 24. That's 7 divisors. Therefore, there are 7 positive integers n > 1 for which 2^24 is a perfect n-th power.

Wait, but hold on. Let me think again. Is there a possibility that for some n, even though n divides 24, there might not be an integer k? But earlier, we saw that if n divides 24, then m = 24/n is an integer, so k = 2^m is an integer. Therefore, for each divisor n > 1 of 24, there exists such a k. Therefore, the answer should be 7.

But let me check with examples. Let's take n=2. Then 2^24 is (2^12)^2, which is 4096^2. That works. For n=3, 2^24 is (2^8)^3, since 8*3=24. 2^8 is 256, so 256^3. That works. For n=4, 2^24 is (2^6)^4, since 6*4=24. 2^6 is 64, so 64^4. That works. For n=6, 2^24 is (2^4)^6, since 4*6=24. 2^4 is 16, so 16^6. That works. For n=8, 2^24 is (2^3)^8, since 3*8=24. 2^3 is 8, so 8^8. That works. For n=12, 2^24 is (2^2)^12, since 2*12=24. 2^2 is 4, so 4^12. That works. For n=24, 2^24 is (2^1)^24, which is 2^24. So that's 2^24. So that works.

So all of these n's work. Therefore, there are 7 such n's.

But wait, let me check if there are any other n's that might work. For example, suppose n is not a divisor of 24. Let's say n=5. Then, can 2^24 be written as k^5? Let's see. If so, then k^5 = 2^24. Then, k would have to be 2^m, so m*5 = 24. But 24 divided by 5 is 4.8, which is not an integer. Therefore, there is no integer m such that m*5=24, so n=5 is not valid. Similarly, n=7: 24 divided by 7 is not integer. So, only divisors of 24 will work. Therefore, the answer is indeed 7.

But let me check if there's a different approach. Maybe considering exponents. Since 2^24 is a perfect n-th power, then the exponent 24 must be divisible by n. Therefore, n must divide 24. Therefore, the number of n>1 is equal to the number of divisors of 24 greater than 1, which is 7.

Alternatively, if I think about the exponents in the prime factorization. For a number to be a perfect n-th power, all exponents in its prime factorization must be multiples of n. In this case, the prime factorization of 2^24 is just 2^24. Therefore, the exponent 24 must be a multiple of n. Therefore, n divides 24. Therefore, same conclusion.

Therefore, the answer is 7.

But let me check once more. The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. Excluding 1, we have 7 numbers. Therefore, the answer is 7.

But wait, the problem says "positive integers n > 1". So, n can be 2,3,4,6,8,12,24. That's 7 numbers. Therefore, the answer is 7.

But let me check if there's a possibility that for some n, even though n divides 24, the number might not be a perfect n-th power? For example, suppose n=2. Then 2^24 is (2^12)^2, which is a perfect square. Similarly, n=3, (2^8)^3. So, yes, all of them are perfect n-th powers.

Alternatively, if n is a divisor of 24, then 24/n is integer, so 2^{24/n} is integer, so (2^{24/n})^n = 2^{24}. Therefore, yes, that works.

Therefore, I think my reasoning is correct. Therefore, the answer is 7.

But wait, hold on. Let me check if n can be something else. For example, suppose n is a composite number. Wait, but n is already a divisor of 24. So, for example, n=4, which is composite. But even if n is composite, as long as it divides 24, it's okay.

Alternatively, maybe there's a case where n is not a divisor of 24, but still 2^24 is a perfect n-th power? For example, suppose n= something like 24/k where k is... Wait, no. If 2^24 is a perfect n-th power, then n must divide 24. Because if you have 2^24 = k^n, then k must be a power of 2, say 2^m, so 2^{24} = 2^{m*n}, so 24 = m*n. Therefore, n divides 24. Therefore, n must be a divisor of 24. Therefore, there are no other possibilities.

Therefore, the answer is 7.

But let me check once again. Let me list all the divisors of 24 greater than 1:

Divisors of 24: 1, 2, 3, 4, 6, 8, 12, 24. So removing 1, we have 2, 3, 4, 6, 8, 12, 24. That's 7 numbers. Therefore, the answer is 7.

Therefore, I think the answer is 7.

But wait, hold on. Wait, the problem says "for how many positive integers n > 1". So, n must be greater than 1. So, 7 numbers.

But let me check if there's a mistake in considering exponents. For example, suppose n= something like 24. Then, 2^24 is (2^1)^24, which is 2^24. So that's a perfect 24th power. Similarly, n=12, 2^24 is (2^2)^12. So, yes.

Alternatively, if n= something like 24/k, but no, n is already a divisor.

Alternatively, maybe there's a case where n is not a divisor of 24, but 2^24 is still a perfect n-th power? For example, suppose n= something like 24 divided by a fraction? But n has to be an integer.

Alternatively, suppose n is a divisor of 24, but maybe there's some other way? For example, if n is  something else. Wait, no. Because if 2^24 is a perfect n-th power, then n must divide 24.

Therefore, I think my answer is correct.

But let me check with another example. Let's take n= something like 24. If n=24, then 2^24 is 2^(24) which is (2^1)^24. So that's a perfect 24th power. Similarly, n=12, 2^24 is (2^2)^12. So, yes.

Alternatively, if n= something like 24/ something. Wait, but n is already a divisor.

Alternatively, maybe n is not a divisor, but 24 is a multiple of n. Wait, but that's the same thing. If n divides 24, then 24 is a multiple of n.

Therefore, I think my answer is correct. Therefore, the number of positive integers n>1 is 7.

But let me check once again. Let me count the divisors again.

Divisors of 24:

Start with 1. Then 2, 3, 4, 6, 8, 12, 24. So that's 8 divisors. Excluding 1, 7 divisors. Therefore, 7 numbers.

Therefore, the answer is 7.

But wait, hold on. Wait, the problem says "perfect n-th power". Is there a case where even if n divides 24, the number might not be a perfect n-th power? For example, suppose n= something. Wait, no. Because if n divides 24, then 24 = n * m, so 2^24 = (2^m)^n. Therefore, it is a perfect n-th power. Therefore, all divisors n>1 of 24 will satisfy the condition.

Therefore, the answer is 7.

But let me check if the problem is maybe considering exponents in a different way. For example, maybe considering that 2^24 is a perfect n-th power if and only if n divides 24. Therefore, the number of n>1 is equal to the number of divisors of 24 greater than 1, which is 7.

Alternatively, maybe there's a mistake in considering that k must be an integer. For example, if n divides 24, then m=24/n is integer, so k=2^m is integer. Therefore, yes.

Therefore, I think my answer is correct.

But just to be thorough, let me check for n=2,3,4,6,8,12,24:

n=2: 2^24 = (2^12)^2. Yes.

n=3: 2^24 = (2^8)^3. Yes.

n=4: 2^24 = (2^6)^4. Yes.

n=6: 2^24 = (2^4)^6. Yes.

n=8: 2^24 = (2^3)^8. Yes.

n=12: 2^24 = (2^2)^12. Yes.

n=24: 2^24 = (2^1)^24. Yes.

All of these work. Therefore, 7 values.

Therefore, the answer is 7.

But wait, hold on. Wait, the problem says "positive integers n>1". So, n=2,3,4,6,8,12,24. That's 7 numbers. Therefore, the answer is 7.

But let me check if there's a different interpretation. For example, maybe the question is asking for how many n>1 such that 2^24 is a perfect n-th power, but maybe considering different exponents? For example, maybe n is not necessarily a divisor of 24? But no, as we saw earlier, if 2^24 is a perfect n-th power, then n must divide 24.

Alternatively, maybe considering exponents in a different way. For example, if 2^24 is a perfect n-th power, then there exists some integer k such that k^n = 2^24. Then, since 2 is prime, k must be a power of 2, so k=2^m, so (2^m)^n = 2^{m*n} = 2^24. Therefore, m*n=24. Therefore, n must divide 24. Therefore, same conclusion.

Therefore, the answer is 7.

Therefore, I think I can be confident that the answer is 7.

**Final Answer**
\boxed{7}

To determine how many positive integers $ n > 1 $ make $ 2^{24} $ a perfect $ n $-th power, we begin by understanding what it means for a number to be a perfect $ n $-th power.

---

### Step 1: Understanding the Definition

A number $ x $ is a **perfect $ n $-th power** if there exists some integer $ k $ such that:

$$
x = k^n
$$

In this case, we are given $ x = 2^{24} $, and we want to find all integers $ n > 1 $ such that:

$$
2^{24} = k^n
$$

---

### Step 2: Prime Factorization Insight

Since $ 2^{24} $ is a power of a prime number, any $ k $ that satisfies $ k^n = 2^{24} $ must also be a power of 2. Let’s suppose:

$$
k = 2^m
$$

Then:

$$
k^n = (2^m)^n = 2^{mn}
$$

So we have:

$$
2^{mn} = 2^{24} \Rightarrow mn = 24
$$

This tells us that $ n $ must be a **positive divisor of 24**, and $ m = \frac{24}{n} $ must also be a positive integer.

---

### Step 3: Counting Valid $ n > 1 $

We now find all **positive divisors of 24** and exclude $ n = 1 $, since the problem specifies $ n > 1 $.

The prime factorization of 24 is:

$$
24 = 2^3 \cdot 3^1
$$

The number of positive divisors is:

$$
(3 + 1)(1 + 1) = 4 \cdot 2 = 8
$$

The divisors of 24 are:

$$
1, 2, 3, 4, 6, 8, 12, 24
$$

Excluding $ n = 1 $, the valid values of $ n $ are:

$$
2, 3, 4, 6, 8, 12, 24
$$

That’s a total of **7** values.

---

### Step 4: Verifying Each Case

For each of these $ n $, we can write:

$$
2^{24} = (2^{24/n})^n
$$

Since $ 24/n $ is an integer for each of these $ n $, the expression is valid, and $ 2^{24} $ is indeed a perfect $ n $-th power.

---

### Final Answer

$$
\boxed{7}
$$

Solution: Note that $2^{24}$ is a perfect $n$th power if and only if $n$ is a divisor of 24. The factors of 24 which are greater than 1 are 2, 3, 4, 6, 8, 12, and 24, so we have $\boxed{7}$ possible values of $n$.