Explanation:
Okay, so I need to factor the quadratic expression $10x^2 - x - 24$ into the form $(Ax - 8)(Bx + 3)$, where A and B are integers. Then, I have to find the value of $AB + B$. Alright, let me think about how to approach this.

First, I remember that factoring a quadratic in the form $ax^2 + bx + c$ usually involves finding two numbers that multiply to $a \times c$ and add up to $b$. But in this case, the problem already gives me a specific form to factor into, so maybe I can use that structure to find A and B.

Let me write down the given expression and the target form:

Given: $10x^2 - x - 24$

Target form: $(Ax - 8)(Bx + 3)$

So, if I expand the target form, I can compare coefficients with the original expression. Let me do that.

Expanding $(Ax - 8)(Bx + 3)$:

First, multiply Ax by Bx: $A \times B \times x^2 = ABx^2$

Then, Ax times 3: $3A x$

Then, -8 times Bx: $-8B x$

Then, -8 times 3: $-24$

So combining all terms:

$ABx^2 + (3A - 8B)x - 24$

Now, this should be equal to the original quadratic $10x^2 - x - 24$. Therefore, the coefficients of corresponding terms must be equal. So:

1. Coefficient of $x^2$: $AB = 10$
2. Coefficient of $x$: $3A - 8B = -1$
3. Constant term: $-24 = -24$ (which checks out)

So, now I have two equations:

1. $AB = 10$
2. $3A - 8B = -1$

I need to find integers A and B that satisfy these equations. Let me think about how to solve this system.

First, since AB = 10, and A and B are integers, I can list the possible integer pairs (A, B) that multiply to 10. Let me recall that 10 can be factored into integers as:

(1, 10), (2, 5), (5, 2), (10, 1), (-1, -10), (-2, -5), (-5, -2), (-10, -1)

So these are all possible pairs. Now, I need to check which of these pairs satisfy the second equation $3A - 8B = -1$.

Let me go through each pair one by one.

First pair: (1, 10)

Check 3(1) - 8(10) = 3 - 80 = -77 ≠ -1. Not good.

Second pair: (2, 5)

3(2) - 8(5) = 6 - 40 = -34 ≠ -1. Nope.

Third pair: (5, 2)

3(5) - 8(2) = 15 - 16 = -1. Oh! That equals -1. So this pair works!

Wait, so A=5 and B=2? Let me check if that's correct.

Let me verify:

AB = 5*2 = 10, which matches the first equation. And 3A -8B = 15 - 16 = -1, which matches the second equation. Perfect!

But let me check the other pairs just to make sure there are no other solutions.

Fourth pair: (10, 1)

3(10) -8(1) = 30 - 8 = 22 ≠ -1. Nope.

Fifth pair: (-1, -10)

3(-1) -8(-10) = -3 + 80 = 77 ≠ -1. Not.

Sixth pair: (-2, -5)

3(-2) -8(-5) = -6 + 40 = 34 ≠ -1. Nope.

Seventh pair: (-5, -2)

3(-5) -8(-2) = -15 + 16 = 1 ≠ -1. Close, but not quite.

Eighth pair: (-10, -1)

3(-10) -8(-1) = -30 + 8 = -22 ≠ -1. Nope.

So only the third pair (5, 2) works. Therefore, A=5 and B=2.

Wait, but let me just make sure that there isn't another way to factor this. Maybe I missed something? Let me check.

Alternatively, maybe I can solve the equations algebraically instead of just trying all possible pairs. Let me try that approach as well to confirm.

We have AB = 10 and 3A - 8B = -1.

From AB = 10, we can express A as 10/B, assuming B ≠ 0. Then substitute into the second equation:

3*(10/B) - 8B = -1

Multiply both sides by B to eliminate the denominator:

3*10 - 8B^2 = -B

Which simplifies to:

30 - 8B² = -B

Bring all terms to one side:

-8B² + B + 30 = 0

Multiply both sides by -1 to make the quadratic coefficient positive:

8B² - B - 30 = 0

Now, solving this quadratic equation for B. Let me use the quadratic formula.

For equation 8B² - B - 30 = 0,

Discriminant D = (-1)^2 - 4*8*(-30) = 1 + 960 = 961

Square root of 961 is 31.

Therefore, B = [1 ± 31]/(2*8) = [1 ± 31]/16

So two solutions:

B = (1 + 31)/16 = 32/16 = 2

B = (1 - 31)/16 = (-30)/16 = -15/8

But since B must be an integer, the second solution is invalid. Therefore, B=2. Then, from AB=10, A=10/B=10/2=5. So A=5, B=2. Same result as before. So that confirms it.

Therefore, A=5 and B=2. Now, the question asks for AB + B.

Compute AB + B. Since AB is 10, and B is 2, so 10 + 2 = 12. Wait, but let me check:

Wait, AB + B = B(A + 1). Alternatively, compute directly:

AB is 5*2=10, then add B which is 2: 10 + 2 = 12. So the answer is 12?

Wait, but let me check again. Maybe I made a mistake here. Let me verify the entire process once more.

Original expression: 10x² -x -24.

Factored form: (5x - 8)(2x + 3). Let me expand this to check.

Multiply 5x by 2x: 10x².

Multiply 5x by 3: 15x.

Multiply -8 by 2x: -16x.

Multiply -8 by 3: -24.

Combine like terms: 10x² + (15x -16x) -24 = 10x² -x -24. Perfect, that's the original expression. So the factoring is correct.

Therefore, A=5, B=2. Then AB + B = (5*2) + 2 = 10 + 2 = 12. So the answer is 12. Wait, but let me check if the problem says AB + B or something else. Let me check again.

The problem says: "What is AB + B?" So yes, AB + B. Which is 10 + 2 = 12. So 12 is the answer. But let me check once more if there's any chance I miscalculated.

Alternatively, maybe the problem expects AB + B to be calculated differently? Let me check.

AB is 5*2=10, and B is 2, so 10 + 2 = 12. Yes. So 12 is correct.

But just to be thorough, let me check if there are other possible factorizations. For example, sometimes quadratics can be factored in multiple ways, but given the specific form (Ax - 8)(Bx + 3), maybe there's only one solution. But let me think.

Suppose someone tried different factors. For example, if I thought of the original quadratic as 10x² -x -24. Let me try factoring it using the standard method.

Looking for two numbers that multiply to (10)*(-24) = -240 and add to -1. Let me think. So factors of -240 that add to -1. Let me think.

Since the product is negative, one number is positive and the other is negative. Their sum is -1. Let me think of factors of 240 that differ by 1. Let me see:

Let me think of pairs of factors of 240:

1 and 240: difference 239

2 and 120: 118

3 and 80: 77

4 and 60: 56

5 and 48: 43

6 and 40: 34

8 and 30: 22

10 and 24: 14

12 and 20: 8

15 and 16: 1

Ah! 15 and 16. Their difference is 1. So if I take 15 and -16, their product is -240 and their sum is -1. Perfect!

So then, we can split the middle term:

10x² - x -24 = 10x² -16x +15x -24

Grouping:

(10x² -16x) + (15x -24)

Factor out common terms:

2x(5x - 8) + 3(5x - 8)

Now, factor out (5x - 8):

(2x + 3)(5x - 8)

Which is the same as (5x - 8)(2x + 3). Which matches our previous result. So indeed, A=5 and B=2. Therefore, AB + B = 10 + 2 = 12. So the answer is 12.

But just to make sure, let me check if there's another way to factor it. For example, if I had chosen different signs. Wait, but since the product of the numbers is -240 and their sum is -1, the only way is 15 and -16. So that's unique. Therefore, the factorization is unique in that way. Therefore, the answer is 12.

Alternatively, maybe I can check if AB + B is 12. Let me think if there's any chance that the problem might have a different answer. Wait, maybe I made a mistake in the sign?

Wait, in the factoring, we have (5x - 8)(2x + 3). So A is 5, B is 2. So AB is 10, and B is 2. So AB + B is 10 + 2 = 12. Yes. So that's correct.

Alternatively, if someone thought that AB + B is A*B + B, which is B*(A + 1). So 2*(5 + 1) = 2*6 = 12. Same result. So that's correct.

Therefore, I think the answer is 12. But let me check once more if there's a different interpretation of the problem.

Wait, the problem says "the expression 10x² -x -24 can be written as (Ax - 8)(Bx + 3), where A and B are integers. What is AB + B?"

So, the question is definitely AB + B. So with A=5, B=2, AB + B = 5*2 + 2 = 12. So yes.

Alternatively, if someone thought that AB + B is A*B + B, which is same as B*(A + 1). So 2*(5 + 1) = 12. Same answer.

Therefore, I think the answer is 12. So I think that's correct.

**Final Answer**
\boxed{12}

To solve the problem, we are given the quadratic expression:

$$
10x^2 - x - 24
$$

and asked to express it in the form:

$$
(Ax - 8)(Bx + 3)
$$

where $ A $ and $ B $ are integers. Then, we are to compute the value of $ AB + B $.

---

### Step 1: Expand the Target Form

We expand the expression $ (Ax - 8)(Bx + 3) $:

$$
(Ax - 8)(Bx + 3) = ABx^2 + 3Ax - 8Bx - 24
$$

Grouping like terms:

$$
ABx^2 + (3A - 8B)x - 24
$$

Now, compare this with the original expression:

$$
10x^2 - x - 24
$$

Matching coefficients:

1. $ AB = 10 $
2. $ 3A - 8B = -1 $
3. Constant term: $ -24 = -24 $ (already matches)

---

### Step 2: Solve the System of Equations

We now solve the system:

$$
AB = 10 \quad \text{(1)} \\
3A - 8B = -1 \quad \text{(2)}
$$

From equation (1), we can express $ A $ in terms of $ B $:

$$
A = \frac{10}{B}
$$

Substitute into equation (2):

$$
3\left(\frac{10}{B}\right) - 8B = -1 \\
\Rightarrow \frac{30}{B} - 8B = -1
$$

Multiply through by $ B $ to eliminate the denominator:

$$
30 - 8B^2 = -B \\
\Rightarrow 8B^2 - B - 30 = 0
$$

Now solve this quadratic equation using the quadratic formula:

$$
B = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(8)(-30)}}{2(8)} = \frac{1 \pm \sqrt{1 + 960}}{16} = \frac{1 \pm \sqrt{961}}{16}
$$

$$
\sqrt{961} = 31 \Rightarrow B = \frac{1 \pm 31}{16}
$$

So:

- $ B = \frac{32}{16} = 2 $
- $ B = \frac{-30}{16} = -\frac{15}{8} $ (not an integer)

Thus, the only integer solution is $ B = 2 $. Then from equation (1):

$$
A = \frac{10}{B} = \frac{10}{2} = 5
$$

---

### Step 3: Compute $ AB + B $

Now that we have $ A = 5 $ and $ B = 2 $, compute:

$$
AB + B = (5 \cdot 2) + 2 = 10 + 2 = \boxed{12}
$$

---

### Final Answer

$$
\boxed{12}
$$

[Assistant]
Explanation:
Okay, so I need to factor the quadratic expression $10x^2 - x - 24$ into the form $(Ax - 8)(Bx + 3)$, where A and B are integers. Then, I have to find the value of $AB + B$. Alright, let me think about how to approach this.

First, I remember that factoring a quadratic in the form $ax^2 + bx + c$ usually involves finding two numbers that multiply to $a \times c$ and add up to $b$. But in this case, the problem already gives me a specific form to factor into, so maybe I can use that structure to find A and B.

Let me write down the given expression and the target form:

Given: $10x^2 - x - 24$

Target form: $(Ax - 8)(Bx + 3)$

So, if I expand the target form, I can compare coefficients with the original expression. Let me do that.

Expanding $(Ax - 8)(Bx + 3)$:

First, multiply Ax by Bx: $A \times B \times x^2 = ABx^2$

Then, Ax times 3: $3A x$

Then, -8 times Bx: $-8B x$

Then, -8 times 3: $-24$

So combining all terms:

$ABx^2 + (3A - 8B)x - 24$

Now, this should be equal to the original quadratic $10x^2 - x - 24$. Therefore, the coefficients of corresponding terms must be equal. So:

1. Coefficient of $x^2$: $AB = 10$
2. Coefficient of $x$: $3A - 8B = -1$
3. Constant term: $-24 = -24$ (which checks out)

So, now I have two equations:

1. $AB = 10$
2. $3A - 8B = -1$

I need to find integers A and B that satisfy these equations. Let me think about how to solve this system.

First, since AB = 10, and A and B are integers, I can list the possible integer pairs (A, B) that multiply to 10. Let me recall that 10 can be factored into integers as:

(1, 10), (2, 5), (5, 2), (10, 1), (-1, -10), (-2, -5), (-5, -2), (-10, -1)

So these are all possible pairs. Now, I need to check which of these pairs satisfy the second equation $3A - 8B = -1$.

Let me go through each pair one by one.

First pair: (1, 10)

Check 3(1) - 8(10) = 3 - 80 = -77 ≠ -1. Not good.

Second pair: (2, 5)

3(2) - 8(5) = 6 - 40 = -34 ≠ -1. Nope.

Third pair: (5, 2)

3(5) - 8(2) = 15 - 16 = -1. Oh! That equals -1. So this pair works!

Wait, so A=5 and B=2? Let me check if that's correct.

Let me verify:

AB = 5*2 = 10, which matches the first equation. And 3A -8B = 15 - 16 = -1, which matches the second equation. Perfect!

But let me check the other pairs just to make sure there are no other solutions.

Fourth pair: (10, 1)

3(10) -8(1) = 30 - 8 = 22 ≠ -1. Nope.

Fifth pair: (-1, -10)

3(-1) -8(-10) = -3 + 80 = 77 ≠ -1. Not.

Sixth pair: (-2, -5)

3(-2) -8(-5) = -6 + 40 = 34 ≠ -1. Nope.

Seventh pair: (-5, -2)

3(-5) -8(-2) = -15 + 16 = 1 ≠ -1. Close, but not quite.

Eighth pair: (-10, -1)

3(-10) -8(-1) = -30 + 8 = -22 ≠ -1. Nope.

So only the third pair (5, 2) works. Therefore, A=5 and B=2.

Wait, but let me just make sure that there isn't another way to factor this. Maybe I missed something? Let me check.

Alternatively, maybe I can solve the equations algebraically instead of just trying all possible pairs. Let me try that approach as well to confirm.

We have AB = 10 and 3A - 8B = -1.

From AB = 10, we can express A as 10/B, assuming B ≠ 0. Then substitute into the second equation:

3*(10/B) - 8B = -1

Multiply both sides by B to eliminate the denominator:

3*10 - 8B^2 = -B

Which simplifies to:

30 - 8B² = -B

Bring all terms to one side:

-8B² + B + 30 = 0

Multiply both sides by -1 to make the quadratic coefficient positive:

8B² - B - 30 = 0

Now, solving this quadratic equation for B. Let me use the quadratic formula.

For equation 8B² - B - 30 = 0,

Discriminant D = (-1)^2 - 4*8*(-30) = 1 + 960 = 961

Square root of 961 is 31.

Therefore, B = [1 ± 31]/(2*8) = [1 ± 31]/16

So two solutions:

B = (1 + 31)/16 = 32/16 = 2

B = (1 - 31)/16 = (-30)/16 = -15/8

But since B must be an integer, the second solution is invalid. Therefore, B=2. Then, from AB=10, A=10/B=10/2=5. So A=5, B=2. Same result as before. So that confirms it.

Therefore, A=5 and B=2. Now, the question asks for AB + B.

Compute AB + B. Since AB is 10, and B is 2, so 10 + 2 = 12. Wait, but let me check:

Wait, AB + B = B(A + 1). Alternatively, compute directly:

AB is 5*2=10, then add B which is 2: 10 + 2 = 12. So the answer is 12?

Wait, but let me check again. Maybe I made a mistake here. Let me verify the entire process once more.

Original expression: 10x² -x -24.

Factored form: (5x - 8)(2x + 3). Let me expand this to check.

Multiply 5x by 2x: 10x².

Multiply 5x by 3: 15x.

Multiply -8 by 2x: -16x.

Multiply -8 by 3: -24.

Combine like terms: 10x² + (15x -16x) -24 = 10x² -x -24. Perfect, that's the original expression. So the factoring is correct.

Therefore, A=5, B=2. Then AB + B = (5*2) + 2 = 10 + 2 = 12. So the answer is 12. Wait, but let me check if the problem says AB + B or something else. Let me check again.

The problem says: "What is AB + B?" So yes, AB + B. Which is 10 + 2 = 12. So 12 is the answer. But let me check once more if there's any chance I miscalculated.

Alternatively, maybe the problem expects AB + B to be calculated differently? Let me check.

AB is 5*2=10, and B is 2, so 10 + 2 = 12. Yes. So 12 is correct.

But just to be thorough, let me check if there are other possible factorizations. For example, sometimes quadratics can be factored in multiple ways, but given the specific form (Ax - 8)(Bx + 3), maybe there's only one solution. But let me think.

Suppose someone tried different factors. For example, if I thought of the original quadratic as 10x² -x -24. Let me try factoring it using the standard method.

Looking for two numbers that multiply to (10)*(-24) = -240 and add to -1. Let me think. So factors of -240 that add to -1. Let me think.

Since the product is negative, one number is positive and the other is negative. Their sum is -1. Let me think of factors of 240 that differ by 1. Let me see:

Let me think of pairs of factors of 240:

1 and 240: difference 239

2 and 120: 118

3 and 80: 77

4 and 60: 56

5 and 48: 43

6 and 40: 34

8 and 30: 22

10 and 24: 14

12 and 20: 8

15 and 16: 1

Ah! 15 and 16. Their difference is 1. So if I take 15 and -16, their product is -240 and their sum is -1. Perfect!

So then, we can split the middle term:

10x² - x -24 = 10x² -16x +15x -24

Grouping:

(10x² -16x) + (15x -24)

Factor out common terms:

2x(5x - 8) + 3(5x - 8)

Now, factor out (5x - 8):

(2x + 3)(5x - 8)

Which is the same as (5x - 8)(2x + 3). Which matches our previous result. So indeed, A=5 and B=2. Therefore, AB + B = 10 + 2 = 12. So the answer is 12.

But just to make sure, let me check if there's another way to factor it. For example, if I had chosen different signs. Wait, but since the product of the numbers is -240 and their sum is -1, the only way is 15 and -16. So that's unique. Therefore, the factorization is unique in that way. Therefore, the answer is 12.

Alternatively, maybe I can check if AB + B is 12. Let me think if there's any chance that the problem might have a different answer. Wait, maybe I made a mistake in the sign?

Wait, in the factoring, we have (5x - 8)(2x + 3). So A is 5, B is 2. So AB is 10, and B is 2. So AB + B is 10 + 2 = 12. Yes. So that's correct.

Alternatively, if someone thought that AB + B is A*B + B, which is B*(A + 1). So 2*(5 + 1) = 2*6 = 12. Same result. So that's correct.

Therefore, I think the answer is 12. But let me check once more if there's a different interpretation of the problem.

Wait, the problem says "the expression 10x² -x -24 can be written as (Ax - 8)(Bx + 3), where A and B are integers. What is AB + B?"

So, the question is definitely AB + B. So with A=5, B=2, AB + B = 5*2 + 2 = 12. So yes.

Alternatively, if someone thought that AB + B is A*B + B, which is same as B*(A + 1). So 2*(5 + 1) = 12. Same answer.

Therefore, I think the answer is 12. So I think that's correct.

**Final Answer**
\boxed{12}

To solve the problem, we are given the quadratic expression:

$$
10x^2 - x - 24
$$

and asked to express it in the form:

$$
(Ax - 8)(Bx + 3)
$$

where $ A $ and $ B $ are integers. Then, we are to compute the value of $ AB + B $.

---

### Step 1: Expand the Target Form

We expand the expression $ (Ax - 8)(Bx + 3) $:

$$
(Ax - 8)(Bx + 3) = ABx^2 + 3Ax - 8Bx - 24
$$

Grouping like terms:

$$
ABx^2 + (3A - 8B)x - 24
$$

Now, compare this with the original expression:

$$
10x^2 - x - 24
$$

Matching coefficients:

1. $ AB = 10 $
2. $ 3A - 8B = -1 $
3. Constant term: $ -24 = -24 $ (already matches)

---

### Step 2: Solve the System of Equations

We now solve the system:

$$
AB = 10 \quad \text{(1)} \\
3A - 8B = -1 \quad \text{(2)}
$$

From equation (1), we can express $ A $ in terms of $ B $:

$$
A = \frac{10}{B}
$$

Substitute into equation (2):

$$
3\left(\frac{10}{B}\right) - 8B = -1 \\
\Rightarrow \frac{30}{B} - 8B = -1
$$

Multiply through by $ B $ to eliminate the denominator:

$$
30 - 8B^2 = -B \\
\Rightarrow 8B^2 - B - 30 = 0
$$

Now solve this quadratic equation using the quadratic formula:

$$
B = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(8)(-30)}}{2(8)} = \frac{1 \pm \sqrt{1 + 960}}{16} = \frac{1 \pm \sqrt{961}}{16}
$$

$$
\sqrt{961} = 31 \Rightarrow B = \frac{1 \pm 31}{16}
$$

So:

- $ B = \frac{32}{16} = 2 $
- $ B = \frac{-30}{16} = -\frac{15}{8} $ (not an integer)

Thus, the only integer solution is $ B = 2 $. Then from equation (1):

$$
A = \frac{10}{B} = \frac{10}{2} = 5
$$

---

### Step 3: Compute $ AB + B $

Now that we have $ A = 5 $ and $ B = 2 $, compute:

$$
AB + B = (5 \cdot 2) + 2 = 10 + 2 = \boxed{12}
$$

---

### Final Answer

$$
\boxed{12}
$$

Solution: We see that $10x^2-x-24=(5x-8)(2x+3)$, thus $A = 5$ and $B = 2$. Hence, $AB + B = \boxed{12}.$


Explanation:
Okay, so I need to combine these two fractions into a single fraction. The expression is (4 + 6a)/5 minus (1 + 3a)/4. Hmm, let me think. To combine fractions, they need a common denominator. The denominators here are 5 and 4. What's the least common denominator for 5 and 4? Well, 5 and 4 are both prime numbers, right? So their least common multiple would be 5 times 4, which is 20. So I need to convert both fractions to have 20 as the denominator.

Starting with the first fraction, (4 + 6a)/5. To get from 5 to 20, I multiply the denominator by 4. So I have to do the same to the numerator. That means multiplying the numerator (4 + 6a) by 4. Let me write that out: 4*(4 + 6a). Let me calculate that. 4 times 4 is 16, and 4 times 6a is 24a. So the first fraction becomes (16 + 24a)/20.

Now the second fraction is (1 + 3a)/4. To convert the denominator from 4 to 20, I need to multiply by 5. So again, I have to multiply the numerator by 5. That would be 5*(1 + 3a). Let me compute that. 5 times 1 is 5, and 5 times 3a is 15a. So the second fraction becomes (5 + 15a)/20.

Now the original expression is the first fraction minus the second fraction, so it's (16 + 24a)/20 minus (5 + 15a)/20. Since the denominators are the same now, I can combine the numerators. So that would be [ (16 + 24a) - (5 + 15a) ] / 20.

Let me simplify the numerator. Distribute the negative sign to both terms in the second numerator. So that becomes 16 + 24a - 5 - 15a. Now combine like terms. Let's do the constants first: 16 - 5 is 11. Then the terms with a: 24a - 15a is 9a. So the numerator simplifies to 11 + 9a. Therefore, the combined fraction is (11 + 9a)/20.

Wait, let me check my steps again to make sure I didn't make a mistake. Starting with the first fraction: 4*(4 + 6a) is indeed 16 + 24a. Second fraction: 5*(1 + 3a) is 5 + 15a. Then subtracting the second fraction from the first: 16 + 24a - 5 - 15a. Yes, 16 - 5 is 11, and 24a - 15a is 9a. So numerator is 11 + 9a over 20. That seems right.

Alternatively, maybe I can write it as (9a + 11)/20. Either way is fine, but maybe they prefer the terms in a certain order. But both are correct. Let me check if the fraction can be simplified further. The numerator is 9a + 11 and the denominator is 20. Are there any common factors between 9, 11, and 20? Well, 9 and 20 have a common factor of 1, 11 and 20 also have a common factor of 1. So the fraction is already in its simplest form.

So the final answer should be (9a + 11)/20. Let me just verify with another approach. Maybe by combining the original expression without finding a common denominator first. Let me try that.

Original expression: (4 + 6a)/5 - (1 + 3a)/4. Let me distribute the division. So that's 4/5 + (6a)/5 - 1/4 - (3a)/4. Then combine like terms. The constants: 4/5 - 1/4. The terms with a: (6a)/5 - (3a)/4. Let me compute each part.

First, constants: 4/5 - 1/4. To subtract these, common denominator is 20 again. 4/5 is 16/20, and 1/4 is 5/20. So 16/20 - 5/20 = 11/20. Then the terms with a: (6a)/5 - (3a)/4. Again, common denominator 20. Multiply numerator and denominator of first term by 4: (6a * 4)/20 = 24a/20. Multiply numerator and denominator of second term by 5: (3a * 5)/20 = 15a/20. So 24a/20 - 15a/20 = 9a/20. Then adding the constant term and the a term: 11/20 + 9a/20 = (9a + 11)/20. Same result. So that confirms it.

Therefore, the expression simplifies to (9a + 11)/20. I think that's correct. Let me check with a sample value for a. Let's pick a = 0. Then original expression: (4 + 0)/5 - (1 + 0)/4 = 4/5 - 1/4. Which is 16/20 - 5/20 = 11/20. The simplified expression: (0 + 11)/20 = 11/20. Same result. Good.

Another test: let a = 1. Original expression: (4 + 6*1)/5 - (1 + 3*1)/4 = (10)/5 - (4)/4 = 2 - 1 = 1. Simplified expression: (9*1 + 11)/20 = (20)/20 = 1. Correct again.

Another test: a = 2. Original expression: (4 + 12)/5 - (1 + 6)/4 = 16/5 - 7/4. Let me compute that. 16/5 is 3.2, 7/4 is 1.75. 3.2 - 1.75 = 1.45. Simplified expression: (9*2 + 11)/20 = (18 + 11)/20 = 29/20 = 1.45. Correct.

So all tests pass. Therefore, the answer is (9a + 11)/20.

**Final Answer**
The expression as a single fraction is \boxed{\dfrac{9a + 11}{20}}.

To simplify the expression
$$
\frac{4+6a}{5} - \frac{1+3a}{4}
$$
into a single fraction, we follow a systematic process of finding a common denominator and combining the numerators.

---

### **Step 1: Identify the Least Common Denominator (LCD)**

The denominators are 5 and 4. Since they are relatively prime, their least common denominator is:

$$
\text{LCD} = 5 \times 4 = 20
$$

---

### **Step 2: Convert Each Fraction to Have the LCD**

We rewrite both fractions with denominator 20:

- For $\frac{4 + 6a}{5}$, multiply numerator and denominator by 4:
  $$
  \frac{4 + 6a}{5} = \frac{4(4 + 6a)}{20} = \frac{16 + 24a}{20}
  $$

- For $\frac{1 + 3a}{4}$, multiply numerator and denominator by 5:
  $$
  \frac{1 + 3a}{4} = \frac{5(1 + 3a)}{20} = \frac{5 + 15a}{20}
  $$

---

### **Step 3: Subtract the Two Fractions**

Now that both fractions have the same denominator, we subtract the numerators:

$$
\frac{16 + 24a}{20} - \frac{5 + 15a}{20} = \frac{(16 + 24a) - (5 + 15a)}{20}
$$

Distribute the negative sign:

$$
= \frac{16 + 24a - 5 - 15a}{20}
$$

Combine like terms:

- Constants: $16 - 5 = 11$
- Terms with $a$: $24a - 15a = 9a$

So the numerator becomes:

$$
11 + 9a
$$

---

### **Step 4: Write the Final Simplified Fraction**

$$
\frac{11 + 9a}{20}
$$

This is the simplified form of the original expression. Since the numerator and denominator have no common factors, the fraction is in its simplest form.

---

### **Final Answer**

$$
\boxed{\dfrac{9a + 11}{20}}
$$

[Assistant]
Explanation:
Okay, so I need to combine these two fractions into a single fraction. The expression is (4 + 6a)/5 minus (1 + 3a)/4. Hmm, let me think. To combine fractions, they need a common denominator. The denominators here are 5 and 4. What's the least common denominator for 5 and 4? Well, 5 and 4 are both prime numbers, right? So their least common multiple would be 5 times 4, which is 20. So I need to convert both fractions to have 20 as the denominator.

Starting with the first fraction, (4 + 6a)/5. To get from 5 to 20, I multiply the denominator by 4. So I have to do the same to the numerator. That means multiplying the numerator (4 + 6a) by 4. Let me write that out: 4*(4 + 6a). Let me calculate that. 4 times 4 is 16, and 4 times 6a is 24a. So the first fraction becomes (16 + 24a)/20.

Now the second fraction is (1 + 3a)/4. To convert the denominator from 4 to 20, I need to multiply by 5. So again, I have to multiply the numerator by 5. That would be 5*(1 + 3a). Let me compute that. 5 times 1 is 5, and 5 times 3a is 15a. So the second fraction becomes (5 + 15a)/20.

Now the original expression is the first fraction minus the second fraction, so it's (16 + 24a)/20 minus (5 + 15a)/20. Since the denominators are the same now, I can combine the numerators. So that would be [ (16 + 24a) - (5 + 15a) ] / 20.

Let me simplify the numerator. Distribute the negative sign to both terms in the second numerator. So that becomes 16 + 24a - 5 - 15a. Now combine like terms. Let's do the constants first: 16 - 5 is 11. Then the terms with a: 24a - 15a is 9a. So the numerator simplifies to 11 + 9a. Therefore, the combined fraction is (11 + 9a)/20.

Wait, let me check my steps again to make sure I didn't make a mistake. Starting with the first fraction: 4*(4 + 6a) is indeed 16 + 24a. Second fraction: 5*(1 + 3a) is 5 + 15a. Then subtracting the second fraction from the first: 16 + 24a - 5 - 15a. Yes, 16 - 5 is 11, and 24a - 15a is 9a. So numerator is 11 + 9a over 20. That seems right.

Alternatively, maybe I can write it as (9a + 11)/20. Either way is fine, but maybe they prefer the terms in a certain order. But both are correct. Let me check if the fraction can be simplified further. The numerator is 9a + 11 and the denominator is 20. Are there any common factors between 9, 11, and 20? Well, 9 and 20 have a common factor of 1, 11 and 20 also have a common factor of 1. So the fraction is already in its simplest form.

So the final answer should be (9a + 11)/20. Let me just verify with another approach. Maybe by combining the original expression without finding a common denominator first. Let me try that.

Original expression: (4 + 6a)/5 - (1 + 3a)/4. Let me distribute the division. So that's 4/5 + (6a)/5 - 1/4 - (3a)/4. Then combine like terms. The constants: 4/5 - 1/4. The terms with a: (6a)/5 - (3a)/4. Let me compute each part.

First, constants: 4/5 - 1/4. To subtract these, common denominator is 20 again. 4/5 is 16/20, and 1/4 is 5/20. So 16/20 - 5/20 = 11/20. Then the terms with a: (6a)/5 - (3a)/4. Again, common denominator 20. Multiply numerator and denominator of first term by 4: (6a * 4)/20 = 24a/20. Multiply numerator and denominator of second term by 5: (3a * 5)/20 = 15a/20. So 24a/20 - 15a/20 = 9a/20. Then adding the constant term and the a term: 11/20 + 9a/20 = (9a + 11)/20. Same result. So that confirms it.

Therefore, the expression simplifies to (9a + 11)/20. I think that's correct. Let me check with a sample value for a. Let's pick a = 0. Then original expression: (4 + 0)/5 - (1 + 0)/4 = 4/5 - 1/4. Which is 16/20 - 5/20 = 11/20. The simplified expression: (0 + 11)/20 = 11/20. Same result. Good.

Another test: let a = 1. Original expression: (4 + 6*1)/5 - (1 + 3*1)/4 = (10)/5 - (4)/4 = 2 - 1 = 1. Simplified expression: (9*1 + 11)/20 = (20)/20 = 1. Correct again.

Another test: a = 2. Original expression: (4 + 12)/5 - (1 + 6)/4 = 16/5 - 7/4. Let me compute that. 16/5 is 3.2, 7/4 is 1.75. 3.2 - 1.75 = 1.45. Simplified expression: (9*2 + 11)/20 = (18 + 11)/20 = 29/20 = 1.45. Correct.

So all tests pass. Therefore, the answer is (9a + 11)/20.

**Final Answer**
The expression as a single fraction is \boxed{\dfrac{9a + 11}{20}}.

To simplify the expression
$$
\frac{4+6a}{5} - \frac{1+3a}{4}
$$
into a single fraction, we follow a systematic process of finding a common denominator and combining the numerators.

---

### **Step 1: Identify the Least Common Denominator (LCD)**

The denominators are 5 and 4. Since they are relatively prime, their least common denominator is:

$$
\text{LCD} = 5 \times 4 = 20
$$

---

### **Step 2: Convert Each Fraction to Have the LCD**

We rewrite both fractions with denominator 20:

- For $\frac{4 + 6a}{5}$, multiply numerator and denominator by 4:
  $$
  \frac{4 + 6a}{5} = \frac{4(4 + 6a)}{20} = \frac{16 + 24a}{20}
  $$

- For $\frac{1 + 3a}{4}$, multiply numerator and denominator by 5:
  $$
  \frac{1 + 3a}{4} = \frac{5(1 + 3a)}{20} = \frac{5 + 15a}{20}
  $$

---

### **Step 3: Subtract the Two Fractions**

Now that both fractions have the same denominator, we subtract the numerators:

$$
\frac{16 + 24a}{20} - \frac{5 + 15a}{20} = \frac{(16 + 24a) - (5 + 15a)}{20}
$$

Distribute the negative sign:

$$
= \frac{16 + 24a - 5 - 15a}{20}
$$

Combine like terms:

- Constants: $16 - 5 = 11$
- Terms with $a$: $24a - 15a = 9a$

So the numerator becomes:

$$
11 + 9a
$$

---

### **Step 4: Write the Final Simplified Fraction**

$$
\frac{11 + 9a}{20}
$$

This is the simplified form of the original expression. Since the numerator and denominator have no common factors, the fraction is in its simplest form.

---

### **Final Answer**

$$
\boxed{\dfrac{9a + 11}{20}}
$$

Solution: The common denominator of $5$ and $4$ is $20$, so we multiply top and bottom of the first fraction by $4$ and multiply top and bottom of the second fraction by $5$. We get \[\frac{4(4+6a)}{4 \cdot 5} - \frac{5(1+3a)}{4 \cdot 5} = \frac{16+24a}{20}-\frac{5+15a}{20}.\] We combine the fractions, being careful to put the numerator of the second fraction in parentheses (as we are subtracting the entire numerator), yielding \[\frac{16+24a-(5+15a)}{20} = \frac{16+24a-5-15a}{20}=\boxed{\frac{11+9a}{20}}.\]


Explanation:
Okay, so I need to figure out how many cylindrical beakers it takes to fill a spherical tank. Let me start by recalling the formulas for the volumes of a cylinder and a sphere.

The volume of a cylinder is given by V_cylinder = πr²h, right? And the volume of a sphere is V_sphere = (4/3)πr³.

Alright, the problem says the beaker is 8 cm high and has a radius of 3 cm. So, first, I should calculate the volume of one beaker. Let me write that down:

V_beaker = π * (3 cm)² * 8 cm. Let me compute that.

First, 3 squared is 9, so that's 9 cm². Then multiply by 8 cm, so 9 * 8 = 72 cm³. Then multiply by π, so the volume of one beaker is 72π cm³.

Now, the spherical tank has a radius of 6 cm. Let me calculate its volume.

V_sphere = (4/3)π*(6 cm)³. Let me compute 6 cubed first. 6*6=36, 36*6=216. So that's 216 cm³. Then multiply by 4/3. Let me do that: 4/3 * 216.

Hmm, 216 divided by 3 is 72, and 72 times 4 is 288. So the volume of the sphere is 288π cm³.

Now, to find out how many beakers are needed to fill the sphere, I need to divide the volume of the sphere by the volume of one beaker.

So number of beakers = V_sphere / V_beaker = 288π / 72π.

Oh, the π cancels out, so that's 288 / 72. Let me compute that. 72 times 4 is 288, right? 72*4=288. So that would be 4.

Wait, so does that mean it takes 4 beakers to fill the spherical tank? That seems straightforward, but let me double-check my calculations to make sure I didn't make a mistake.

First, volume of the cylinder: radius 3 cm, height 8 cm. Area of the base is πr² = π*9, times height 8, so 72π. Correct.

Volume of the sphere: radius 6 cm. The formula is (4/3)πr³. So (4/3)*π*216. 216 divided by 3 is 72, times 4 is 288. So 288π. Correct.

Dividing 288π by 72π gives 4. So yes, 4 beakers.

Wait, but intuitively, a sphere and a cylinder... Is there something about the shape that might affect this? Like, maybe if you're pouring water from cylinders into a sphere, there might be some inefficiency? But the question says "how many such beakers of water will it take to fill a spherical tank," so I think it's just a matter of volume. Since volume is a measure of capacity, regardless of the shape, the number of beakers needed would just be the ratio of the volumes.

So unless there's some trick here, like maybe the beakers can't be arranged perfectly or something? But the problem doesn't mention anything like that. It just asks for the number based on volume.

So I think my answer is correct. 4 beakers.

But let me think again. Maybe I miscalculated the volumes? Let me check once more.

Cylinder: radius 3, height 8. Volume is πr²h = π*9*8 = 72π. Sphere: radius 6. Volume is (4/3)π*(6)^3. 6^3 is 216. 4/3 of 216: 216 divided by 3 is 72, times 4 is 288. So sphere is 288π. Divided by 72π is 4.

Yes, that seems right.

Alternatively, maybe the question is trying to trick me into thinking about something else? For example, if the beakers are filled to the brim and poured into the sphere, but since the sphere is a different shape, maybe there's some overlap or something? But no, volume-wise, it's just the total volume.

Alternatively, maybe the beakers are not being filled completely? But the question says "how many such beakers of water will it take to fill," so I think it's assuming that each beaker is completely filled and poured into the tank.

Therefore, I think the answer is 4.

But just to be thorough, let me think of another way. Let me compute the numerical values without π.

Volume of beaker: 72π. Volume of sphere: 288π. So ratio is 288/72 = 4.

Alternatively, if I compute the numerical values with π, but since π cancels, it's still 4.

Alternatively, maybe the question is in different units? But no, both are in cm.

Wait, the beaker is 8 cm high, radius 3 cm. Sphere radius 6 cm. So the sphere has a radius twice that of the beaker's radius, and the height of the beaker is 8 cm.

But maybe there's a relation between the dimensions? For example, if you consider the sphere's diameter is 12 cm, which is twice the radius. The beaker's height is 8 cm, which is less than the sphere's diameter.

But I don't think that affects the volume calculation.

Alternatively, maybe the question is expecting me to consider that the sphere can't be filled perfectly with cylinders? But again, unless there's some packing efficiency, which is not mentioned here, the answer should be purely based on volume.

So, I think my answer is correct. 4 beakers.

**Final Answer**
The number of beakers needed is \boxed{4}.

To determine how many cylindrical beakers are required to fill a spherical tank, we begin by calculating the volumes of both the beaker and the tank using their respective geometric formulas.

---

### Step 1: Volume of a Cylindrical Beaker

The formula for the volume of a cylinder is:

$$
V_{\text{cylinder}} = \pi r^2 h
$$

Given:
- Radius $ r = 3 \, \text{cm} $
- Height $ h = 8 \, \text{cm} $

$$
V_{\text{beaker}} = \pi (3)^2 (8) = \pi \cdot 9 \cdot 8 = 72\pi \, \text{cm}^3
$$

---

### Step 2: Volume of the Spherical Tank

The formula for the volume of a sphere is:

$$
V_{\text{sphere}} = \frac{4}{3} \pi r^3
$$

Given:
- Radius $ r = 6 \, \text{cm} $

$$
V_{\text{sphere}} = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi \cdot 216 = 288\pi \, \text{cm}^3
$$

---

### Step 3: Determine the Number of Beakers Needed

To find how many beakers are required to fill the spherical tank, divide the volume of the sphere by the volume of one beaker:

$$
\text{Number of beakers} = \frac{V_{\text{sphere}}}{V_{\text{beaker}}} = \frac{288\pi}{72\pi} = \frac{288}{72} = 4
$$

---

### Final Answer

$$
\boxed{4}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many cylindrical beakers it takes to fill a spherical tank. Let me start by recalling the formulas for the volumes of a cylinder and a sphere.

The volume of a cylinder is given by V_cylinder = πr²h, right? And the volume of a sphere is V_sphere = (4/3)πr³.

Alright, the problem says the beaker is 8 cm high and has a radius of 3 cm. So, first, I should calculate the volume of one beaker. Let me write that down:

V_beaker = π * (3 cm)² * 8 cm. Let me compute that.

First, 3 squared is 9, so that's 9 cm². Then multiply by 8 cm, so 9 * 8 = 72 cm³. Then multiply by π, so the volume of one beaker is 72π cm³.

Now, the spherical tank has a radius of 6 cm. Let me calculate its volume.

V_sphere = (4/3)π*(6 cm)³. Let me compute 6 cubed first. 6*6=36, 36*6=216. So that's 216 cm³. Then multiply by 4/3. Let me do that: 4/3 * 216.

Hmm, 216 divided by 3 is 72, and 72 times 4 is 288. So the volume of the sphere is 288π cm³.

Now, to find out how many beakers are needed to fill the sphere, I need to divide the volume of the sphere by the volume of one beaker.

So number of beakers = V_sphere / V_beaker = 288π / 72π.

Oh, the π cancels out, so that's 288 / 72. Let me compute that. 72 times 4 is 288, right? 72*4=288. So that would be 4.

Wait, so does that mean it takes 4 beakers to fill the spherical tank? That seems straightforward, but let me double-check my calculations to make sure I didn't make a mistake.

First, volume of the cylinder: radius 3 cm, height 8 cm. Area of the base is πr² = π*9, times height 8, so 72π. Correct.

Volume of the sphere: radius 6 cm. The formula is (4/3)πr³. So (4/3)*π*216. 216 divided by 3 is 72, times 4 is 288. So 288π. Correct.

Dividing 288π by 72π gives 4. So yes, 4 beakers.

Wait, but intuitively, a sphere and a cylinder... Is there something about the shape that might affect this? Like, maybe if you're pouring water from cylinders into a sphere, there might be some inefficiency? But the question says "how many such beakers of water will it take to fill a spherical tank," so I think it's just a matter of volume. Since volume is a measure of capacity, regardless of the shape, the number of beakers needed would just be the ratio of the volumes.

So unless there's some trick here, like maybe the beakers can't be arranged perfectly or something? But the problem doesn't mention anything like that. It just asks for the number based on volume.

So I think my answer is correct. 4 beakers.

But let me think again. Maybe I miscalculated the volumes? Let me check once more.

Cylinder: radius 3, height 8. Volume is πr²h = π*9*8 = 72π. Sphere: radius 6. Volume is (4/3)π*(6)^3. 6^3 is 216. 4/3 of 216: 216 divided by 3 is 72, times 4 is 288. So sphere is 288π. Divided by 72π is 4.

Yes, that seems right.

Alternatively, maybe the question is trying to trick me into thinking about something else? For example, if the beakers are filled to the brim and poured into the sphere, but since the sphere is a different shape, maybe there's some overlap or something? But no, volume-wise, it's just the total volume.

Alternatively, maybe the beakers are not being filled completely? But the question says "how many such beakers of water will it take to fill," so I think it's assuming that each beaker is completely filled and poured into the tank.

Therefore, I think the answer is 4.

But just to be thorough, let me think of another way. Let me compute the numerical values without π.

Volume of beaker: 72π. Volume of sphere: 288π. So ratio is 288/72 = 4.

Alternatively, if I compute the numerical values with π, but since π cancels, it's still 4.

Alternatively, maybe the question is in different units? But no, both are in cm.

Wait, the beaker is 8 cm high, radius 3 cm. Sphere radius 6 cm. So the sphere has a radius twice that of the beaker's radius, and the height of the beaker is 8 cm.

But maybe there's a relation between the dimensions? For example, if you consider the sphere's diameter is 12 cm, which is twice the radius. The beaker's height is 8 cm, which is less than the sphere's diameter.

But I don't think that affects the volume calculation.

Alternatively, maybe the question is expecting me to consider that the sphere can't be filled perfectly with cylinders? But again, unless there's some packing efficiency, which is not mentioned here, the answer should be purely based on volume.

So, I think my answer is correct. 4 beakers.

**Final Answer**
The number of beakers needed is \boxed{4}.

To determine how many cylindrical beakers are required to fill a spherical tank, we begin by calculating the volumes of both the beaker and the tank using their respective geometric formulas.

---

### Step 1: Volume of a Cylindrical Beaker

The formula for the volume of a cylinder is:

$$
V_{\text{cylinder}} = \pi r^2 h
$$

Given:
- Radius $ r = 3 \, \text{cm} $
- Height $ h = 8 \, \text{cm} $

$$
V_{\text{beaker}} = \pi (3)^2 (8) = \pi \cdot 9 \cdot 8 = 72\pi \, \text{cm}^3
$$

---

### Step 2: Volume of the Spherical Tank

The formula for the volume of a sphere is:

$$
V_{\text{sphere}} = \frac{4}{3} \pi r^3
$$

Given:
- Radius $ r = 6 \, \text{cm} $

$$
V_{\text{sphere}} = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi \cdot 216 = 288\pi \, \text{cm}^3
$$

---

### Step 3: Determine the Number of Beakers Needed

To find how many beakers are required to fill the spherical tank, divide the volume of the sphere by the volume of one beaker:

$$
\text{Number of beakers} = \frac{V_{\text{sphere}}}{V_{\text{beaker}}} = \frac{288\pi}{72\pi} = \frac{288}{72} = 4
$$

---

### Final Answer

$$
\boxed{4}
$$

Solution: We first have to remember our formulas for the volumes of 3 dimensional objects. The volume of a cylinder with radius $r$ and height $h$ is $r^2h\pi$ and the volume of a sphere with radius $r$ is $\frac{4}{3} r^3 \pi$. Since the cylindrical beaker has a height of 8 centimeters and a radius of 3 centimeters, that means that its volume is $3^2\cdot8\cdot\pi=72\pi$ cubic centimeters. Since the sphere has a radius of 6 centimeters, its volume is $\frac{4}{3}\cdot6^3\pi = 288\pi$ cubic centimeters. The number of beakers of what it will take to fill the spherical tank is just the ratio of the volume of the tank to the volume of the cylinder, which is given by $\dfrac{288\pi}{72\pi}=\boxed{4}$.


Explanation:
Okay, so I need to evaluate this expression: sin(arcsin 0.4 + arcsin 0.5) multiplied by sin(arcsin 0.5 - arcsin 0.4). Hmm, that seems a bit complicated at first glance, but maybe I can break it down using some trigonometric identities. Let me recall... there are formulas for sine of sum and difference of angles. Let me write them down.

The sine of a sum formula is sin(A + B) = sin A cos B + cos A sin B, and the sine of a difference is sin(A - B) = sin A cos B - cos A sin B. So maybe I can apply these formulas here. Let me denote arcsin 0.4 as α and arcsin 0.5 as β. Then the expression becomes sin(α + β) * sin(β - α). Wait, that's interesting. So the original expression is sin(α + β) multiplied by sin(β - α).

Alternatively, maybe I can compute each part separately. Let me first compute α = arcsin 0.4 and β = arcsin 0.5. Then compute sin(α + β) and sin(β - α), then multiply them. But maybe there's a smarter way. Let me think.

Alternatively, since the expression is sin(α + β) * sin(β - α), maybe I can use some identity to combine these terms. Let me recall that sin A sin B can be expressed as [cos(A - B) - cos(A + B)] / 2. Wait, is that right? Let me check. The product-to-sum formulas: yes, sin A sin B = [cos(A - B) - cos(A + B)] / 2. So if I let A = α + β and B = β - α, then the product would be [cos((α + β) - (β - α)) - cos((α + β) + (β - α))]/2. Let me compute those arguments.

First, (α + β) - (β - α) = α + β - β + α = 2α. Then, (α + β) + (β - α) = α + β + β - α = 2β. Therefore, the product becomes [cos(2α) - cos(2β)] / 2. So the original expression simplifies to [cos(2α) - cos(2β)] / 2.

Hmm, that seems simpler. So instead of computing sin(α + β) and sin(β - α) separately, I can compute cos(2α) and cos(2β) and then take their difference. Let me check if this is correct. Let me verify with another approach.

Alternatively, if I compute sin(α + β) * sin(β - α), maybe I can also think of it as sin(α + β) * sin(β - α) = [sin β cos α + cos β sin α] * [sin β cos α - cos β sin α]. Wait, that's like (x + y)(x - y) where x = sin β cos α and y = cos β sin α. Therefore, this would be x² - y². So expanding that, it's (sin β cos α)^2 - (cos β sin α)^2.

Let me compute that. So sin² β cos² α - cos² β sin² α. Hmm, maybe I can factor this expression. Let me see. Alternatively, maybe I can use the identity for cos(2α) and cos(2β). Let me recall that cos(2θ) = 1 - 2 sin² θ, or 2 cos² θ - 1. Also, cos(2θ) = cos² θ - sin² θ. Wait, so if I take cos(2α) - cos(2β) = [cos² α - sin² α] - [cos² β - sin² β] = (cos² α - cos² β) - (sin² α - sin² β). Hmm, not sure if that helps.

But earlier, I had that the product sin(α + β) * sin(β - α) equals [cos(2α) - cos(2β)] / 2. Let me check if that's consistent with the other approach. Let me compute [cos(2α) - cos(2β)] / 2. If I use the identity for cos(2α) - cos(2β), which is -2 sin(α + β) sin(α - β). Wait, there's another identity: cos A - cos B = -2 sin[(A + B)/2] sin[(A - B)/2]. So cos(2α) - cos(2β) = -2 sin[(2α + 2β)/2] sin[(2α - 2β)/2] = -2 sin(α + β) sin(α - β). Therefore, [cos(2α) - cos(2β)] / 2 = - sin(α + β) sin(α - β). But the original expression is sin(α + β) * sin(β - α). Note that sin(β - α) = - sin(α - β). Therefore, sin(α + β) * sin(β - α) = - sin(α + β) sin(α - β). Which matches the previous result. Therefore, the original expression is equal to [cos(2α) - cos(2β)] / 2.

So either way, I can compute it as [cos(2α) - cos(2β)] / 2. Maybe this is easier? Let me try that.

So first, I need to compute cos(2α) and cos(2β). Since α = arcsin 0.4, then sin α = 0.4. Similarly, sin β = 0.5.

To find cos(2α), I can use the identity cos(2θ) = 1 - 2 sin² θ. Similarly, cos(2β) = 1 - 2 sin² β.

Alternatively, cos(2θ) = 2 cos² θ - 1. Either way, I need to find cos α and cos β.

Since α = arcsin 0.4, then cos α = sqrt(1 - sin² α) = sqrt(1 - 0.16) = sqrt(0.84). Similarly, cos β = sqrt(1 - sin² β) = sqrt(1 - 0.25) = sqrt(0.75).

But wait, arcsin returns values between -π/2 and π/2, so cos α and cos β are positive. Therefore, cos α = sqrt(1 - 0.16) = sqrt(0.84). Let me compute sqrt(0.84). Well, 0.84 is 84/100 = 21/25. Wait, sqrt(21)/5? Wait, sqrt(21)/5 squared is 21/25 = 0.84. Yes. So sqrt(0.84) is sqrt(21)/5. Similarly, sqrt(0.75) is sqrt(3)/2.

Therefore, cos α = sqrt(21)/5 and cos β = sqrt(3)/2.

So let me compute cos(2α) using the identity cos(2θ) = 1 - 2 sin² θ. For cos(2α): 1 - 2*(0.4)^2 = 1 - 2*(0.16) = 1 - 0.32 = 0.68. Alternatively, using cos(2α) = 2 cos² α - 1. Let me check that. 2*(sqrt(21)/5)^2 - 1 = 2*(21/25) - 1 = 42/25 - 1 = 17/25 = 0.68. Yes, same result.

Similarly, cos(2β) = 1 - 2*(0.5)^2 = 1 - 2*(0.25) = 1 - 0.5 = 0.5. Alternatively, 2*(sqrt(3)/2)^2 - 1 = 2*(3/4) - 1 = 3/2 - 1 = 1/2. Correct.

Therefore, cos(2α) - cos(2β) = 0.68 - 0.5 = 0.18. Then divide by 2: 0.18 / 2 = 0.09. Therefore, the original expression is 0.09.

Wait, but let me check if that's correct. Let me verify with another approach. Let me compute sin(α + β) and sin(β - α) separately and multiply them.

First, compute α = arcsin(0.4). Let me find sin(α + β). Using the formula sin(A + B) = sin A cos B + cos A sin B. So sin α = 0.4, cos α = sqrt(21)/5 ≈ sqrt(21)/5 ≈ 4.583/5 ≈ 0.9166. Wait, sqrt(21) is approx 4.58366, so divided by 5 is approx 0.9166. Similarly, sin β = 0.5, cos β = sqrt(3)/2 ≈ 0.8660.

Therefore, sin(α + β) = sin α cos β + cos α sin β = 0.4*(sqrt(3)/2) + (sqrt(21)/5)*0.5. Let me compute that.

First term: 0.4*(sqrt(3)/2) = 0.2*sqrt(3) ≈ 0.2*1.732 ≈ 0.3464.

Second term: (sqrt(21)/5)*0.5 = sqrt(21)/10 ≈ 4.583/10 ≈ 0.4583.

Adding them together: approx 0.3464 + 0.4583 ≈ 0.8047.

Now compute sin(β - α) = sin β cos α - cos β sin α.

Which is 0.5*(sqrt(21)/5) - (sqrt(3)/2)*0.4.

First term: 0.5*(sqrt(21)/5) = sqrt(21)/10 ≈ 0.4583.

Second term: (sqrt(3)/2)*0.4 = 0.4*(sqrt(3)/2) = 0.2*sqrt(3) ≈ 0.3464.

Therefore, sin(β - α) ≈ 0.4583 - 0.3464 ≈ 0.1119.

Then multiplying sin(α + β) * sin(β - α) ≈ 0.8047 * 0.1119 ≈ Let me compute that.

First, 0.8 * 0.1119 ≈ 0.0895, and 0.0047 * 0.1119 ≈ approx 0.000526. So total approx 0.0895 + 0.0005 ≈ 0.09. Which matches the previous result of 0.09.

So that seems to confirm that the answer is 0.09. But let me check if there's an exact value.

Earlier, we had [cos(2α) - cos(2β)] / 2 = (0.68 - 0.5)/2 = 0.18 / 2 = 0.09. But 0.68 is 17/25, and 0.5 is 1/2. Therefore, 17/25 - 1/2 = (34 - 25)/50 = 9/50. Then divided by 2 is 9/100, which is 0.09. So exact value is 9/100.

Therefore, the answer is 9/100, which is 0.09.

Alternatively, maybe I can check using exact expressions. Let me try that.

We had cos(2α) = 17/25 and cos(2β) = 1/2. Therefore, cos(2α) - cos(2β) = 17/25 - 1/2 = (34 - 25)/50 = 9/50. Then divided by 2 is 9/100. So yes, exactly 9/100.

Alternatively, if I compute sin(α + β) * sin(β - α) using exact terms.

Earlier, we had sin(α + β) = 0.4*(sqrt(3)/2) + (sqrt(21)/5)*0.5. Let me write that as fractions.

0.4 is 2/5, so 2/5 * sqrt(3)/2 = (2/5)*(sqrt(3)/2) = sqrt(3)/5.

Similarly, sqrt(21)/5 * 0.5 is sqrt(21)/5 * 1/2 = sqrt(21)/10.

Therefore, sin(α + β) = sqrt(3)/5 + sqrt(21)/10.

Similarly, sin(β - α) = sqrt(21)/10 - sqrt(3)/5.

Therefore, multiplying them: [sqrt(3)/5 + sqrt(21)/10] * [sqrt(21)/10 - sqrt(3)/5].

This is of the form (a + b)(b - a) = b² - a², where a = sqrt(3)/5 and b = sqrt(21)/10.

Wait, let me check. Let me denote a = sqrt(3)/5 and b = sqrt(21)/10. Then the expression is (a + b)(b - a) = b² - a².

Compute b²: (sqrt(21)/10)^2 = 21/100.

Compute a²: (sqrt(3)/5)^2 = 3/25 = 12/100.

Therefore, b² - a² = 21/100 - 12/100 = 9/100.

Exactly! Therefore, the product is 9/100.

So that's the exact value. Therefore, the answer is 9/100.

Therefore, regardless of the method, I get 9/100.

Alternatively, maybe there's another way to think about this problem. Let me think.

Suppose I let x = arcsin 0.4 and y = arcsin 0.5. Then the problem is sin(x + y) * sin(y - x).

Alternatively, note that sin(x + y) * sin(y - x) = [sin y cos x + cos y sin x][sin y cos x - cos y sin x] = (sin y cos x)^2 - (cos y sin x)^2.

Which is sin² y cos² x - cos² y sin² x.

Factor this expression. Let me see.

sin² y cos² x - cos² y sin² x = sin² y cos² x - cos² y sin² x.

Hmm, maybe factor sin² y cos² x - cos² y sin² x. Let me factor sin² y cos² x - cos² y sin² x.

Alternatively, factor out sin² y cos² x - cos² y sin² x = sin² y cos² x - cos² y sin² x.

Let me write this as sin² y cos² x - cos² y sin² x = sin² y cos² x - sin² x cos² y.

Which is equal to sin² y cos² x - sin² x cos² y.

This can be written as (sin y cos x - sin x cos y)(sin y cos x + sin x cos y). Wait, but that's the same as sin(y - x) * sin(y + x). Which brings us back to the original expression. So that's circular.

Alternatively, maybe express in terms of sin(y - x) and sin(y + x). But we already did that.

Alternatively, maybe use the identity sin² A - sin² B = sin(A - B) sin(A + B). Wait, yes! Because sin² A - sin² B = (sin A - sin B)(sin A + sin B). But also, using the identity sin² A - sin² B = sin(A - B) sin(A + B). Let me check that.

Yes, because sin(A - B) sin(A + B) = [sin A cos B - cos A sin B][sin A cos B + cos A sin B] = sin² A cos² B - cos² A sin² B. Which is not exactly the same as sin² A - sin² B. Wait, unless we manipulate further.

Alternatively, let me recall that sin² A - sin² B = (sin A - sin B)(sin A + sin B). But also, using the identity sin² A - sin² B = (sin A - sin B)(sin A + sin B) = [2 cos((A + B)/2) sin((A - B)/2)][2 sin((A + B)/2) cos((A - B)/2)] = 4 sin((A + B)/2) cos((A + B)/2) sin((A - B)/2) cos((A - B)/2) = sin(A + B) sin(A - B).

Yes! Therefore, sin² A - sin² B = sin(A + B) sin(A - B). Therefore, in our case, sin² y cos² x - cos² y sin² x = sin² y cos² x - sin² x cos² y = [sin y cos x]^2 - [sin x cos y]^2 = sin² y cos² x - sin² x cos² y = sin(y + x) sin(y - x). Wait, but that's the original expression. Wait, no. Wait, according to the identity, sin² A - sin² B = sin(A + B) sin(A - B). But here, we have [sin y cos x]^2 - [sin x cos y]^2. Let me think of this as sin² y cos² x - sin² x cos² y. Let me factor this as cos² x sin² y - cos² y sin² x.

Let me factor out sin² y cos² x - sin² x cos² y. Let me write this as sin² y cos² x - sin² x cos² y = sin² y cos² x - sin² x cos² y.

Let me factor this expression. Let me think of it as (sin y cos x - sin x cos y)(sin y cos x + sin x cos y). Which is sin(y - x) * sin(y + x). Which is the original expression. Therefore, again, circular.

But earlier, we found that this expression equals 9/100.

Alternatively, maybe I can compute sin(α + β) and sin(β - α) numerically with more precision. Let me check.

First, compute α = arcsin(0.4). Let me compute α in radians.

Well, arcsin(0.4) is approximately... Let me recall that sin(0.4115 radians) ≈ 0.4. Because sin(0.4115) ≈ 0.4. Let me check with calculator.

But since I don't have a calculator here, but maybe I can use known values. For example, sin(π/6) = 0.5, which is 0.5. So 0.4 is less than that. π/6 is approx 0.5236 radians. So arcsin(0.4) is less than π/6. Let me approximate.

Alternatively, maybe use Taylor series? Not sure. Alternatively, maybe just accept that we can compute the exact value as 9/100.

But since we already have two methods giving 9/100, and the approximate decimal calculation also gives 0.09, which is 9/100, I think that's solid.

Therefore, the answer is 9/100.

But just to make sure, let me check with another approach. Let me compute the numerical values of α and β, then compute the angles, then compute the sines.

First, compute α = arcsin(0.4). Let me use a calculator (pretending I have one). Let me recall that arcsin(0.4) ≈ 0.4115 radians. Similarly, arcsin(0.5) is π/6 ≈ 0.5236 radians.

Therefore, α + β ≈ 0.4115 + 0.5236 ≈ 0.9351 radians. Then sin(0.9351) ≈ sin(0.9351). Let me recall that sin(π/3) ≈ 0.8660, π/3 ≈ 1.0472 radians. So 0.9351 is less than π/3. Let me approximate sin(0.9351).

Alternatively, use the value we calculated earlier: approx 0.8047. Let me check with calculator steps.

Alternatively, use the exact expression: sin(α + β) = sqrt(3)/5 + sqrt(21)/10 ≈ 1.732/5 + 4.583/10 ≈ 0.3464 + 0.4583 ≈ 0.8047.

Similarly, sin(β - α) ≈ 0.1119.

Multiplying them gives approx 0.8047 * 0.1119 ≈ 0.09.

Alternatively, if I compute 0.8047 * 0.1119:

First, 0.8 * 0.1119 = 0.08952

Then 0.0047 * 0.1119 ≈ 0.000526

Total ≈ 0.08952 + 0.000526 ≈ 0.090046, which is approx 0.09.

Therefore, 0.09 is accurate.

Therefore, the exact value is 9/100, which is 0.09.

Therefore, the answer is 9/100.

But let me check once more with another identity.

We had earlier that the product sin(α + β) * sin(β - α) equals [cos(2α) - cos(2β)] / 2.

We found cos(2α) = 17/25 and cos(2β) = 1/2. Therefore, [17/25 - 1/2]/2 = [ (34 - 25)/50 ] / 2 = (9/50)/2 = 9/100.

Yes, that's correct.

Alternatively, if I use the formula for sin(A)sin(B) = [cos(A - B) - cos(A + B)] / 2.

Wait, in our case, A = α + β and B = β - α. Therefore, sin(A)sin(B) = [cos(A - B) - cos(A + B)] / 2.

Compute A - B = (α + β) - (β - α) = 2α.

A + B = (α + β) + (β - α) = 2β.

Therefore, sin(A)sin(B) = [cos(2α) - cos(2β)] / 2. Which is exactly what we had before.

Therefore, this confirms again that the value is 9/100.

Therefore, I think I can be confident that the answer is 9/100.

But just to make sure, let me check if there's any possible mistake in the calculation of cos(2α) and cos(2β).

For α = arcsin(0.4), sin α = 0.4. Therefore, cos α = sqrt(1 - 0.16) = sqrt(0.84). Then cos(2α) = 1 - 2*(0.4)^2 = 1 - 0.32 = 0.68. Which is 17/25. Because 0.68 = 68/100 = 17/25. Correct.

Similarly, for β = arcsin(0.5), sin β = 0.5, so cos β = sqrt(1 - 0.25) = sqrt(0.75) = sqrt(3)/2. Then cos(2β) = 1 - 2*(0.5)^2 = 1 - 0.5 = 0.5 = 1/2. Correct.

Therefore, cos(2α) - cos(2β) = 17/25 - 1/2 = 34/50 - 25/50 = 9/50. Divided by 2 is 9/100.

Yes, that's correct.

Therefore, the answer is 9/100.

Alternatively, if I convert 9/100 to decimal, it's 0.09, which matches the approximate calculation.

Therefore, I think this is solid.

So, the final answer is 9/100, which can be written as \boxed{\dfrac{9}{100}}.

**Final Answer**
\boxed{\dfrac{9}{100}}

To evaluate the expression:

$$
\sin (\arcsin 0.4 + \arcsin 0.5) \cdot \sin (\arcsin 0.5 - \arcsin 0.4),
$$

we begin by introducing substitutions to simplify the notation. Let:

- $ \alpha = \arcsin 0.4 $, so $ \sin \alpha = 0.4 $,
- $ \beta = \arcsin 0.5 $, so $ \sin \beta = 0.5 $.

Then the expression becomes:

$$
\sin(\alpha + \beta) \cdot \sin(\beta - \alpha).
$$

---

### Step 1: Use Trigonometric Identities

We apply the **product-to-sum identity** for sine:

$$
\sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)].
$$

Let $ A = \alpha + \beta $ and $ B = \beta - \alpha $. Then:

$$
\sin(\alpha + \beta) \cdot \sin(\beta - \alpha) = \frac{1}{2} [\cos((\alpha + \beta) - (\beta - \alpha)) - \cos((\alpha + \beta) + (\beta - \alpha))].
$$

Simplify the arguments:

- $ (\alpha + \beta) - (\beta - \alpha) = 2\alpha $,
- $ (\alpha + \beta) + (\beta - \alpha) = 2\beta $.

So the expression becomes:

$$
\frac{1}{2} [\cos(2\alpha) - \cos(2\beta)].
$$

---

### Step 2: Compute $ \cos(2\alpha) $ and $ \cos(2\beta) $

We use the identity:

$$
\cos(2\theta) = 1 - 2\sin^2 \theta.
$$

- For $ \alpha $, $ \sin \alpha = 0.4 $, so:
  $$
  \cos(2\alpha) = 1 - 2(0.4)^2 = 1 - 2(0.16) = 1 - 0.32 = 0.68.
  $$

- For $ \beta $, $ \sin \beta = 0.5 $, so:
  $$
  \cos(2\beta) = 1 - 2(0.5)^2 = 1 - 2(0.25) = 1 - 0.5 = 0.5.
  $$

---

### Step 3: Plug into the Expression

Now compute:

$$
\frac{1}{2} [\cos(2\alpha) - \cos(2\beta)] = \frac{1}{2} [0.68 - 0.5] = \frac{1}{2} \cdot 0.18 = 0.09.
$$

---

### Step 4: Express as a Fraction

Since $ 0.09 = \frac{9}{100} $, the exact value of the expression is:

$$
\boxed{\dfrac{9}{100}}
$$

[Assistant]
Explanation:
Okay, so I need to evaluate this expression: sin(arcsin 0.4 + arcsin 0.5) multiplied by sin(arcsin 0.5 - arcsin 0.4). Hmm, that seems a bit complicated at first glance, but maybe I can break it down using some trigonometric identities. Let me recall... there are formulas for sine of sum and difference of angles. Let me write them down.

The sine of a sum formula is sin(A + B) = sin A cos B + cos A sin B, and the sine of a difference is sin(A - B) = sin A cos B - cos A sin B. So maybe I can apply these formulas here. Let me denote arcsin 0.4 as α and arcsin 0.5 as β. Then the expression becomes sin(α + β) * sin(β - α). Wait, that's interesting. So the original expression is sin(α + β) multiplied by sin(β - α).

Alternatively, maybe I can compute each part separately. Let me first compute α = arcsin 0.4 and β = arcsin 0.5. Then compute sin(α + β) and sin(β - α), then multiply them. But maybe there's a smarter way. Let me think.

Alternatively, since the expression is sin(α + β) * sin(β - α), maybe I can use some identity to combine these terms. Let me recall that sin A sin B can be expressed as [cos(A - B) - cos(A + B)] / 2. Wait, is that right? Let me check. The product-to-sum formulas: yes, sin A sin B = [cos(A - B) - cos(A + B)] / 2. So if I let A = α + β and B = β - α, then the product would be [cos((α + β) - (β - α)) - cos((α + β) + (β - α))]/2. Let me compute those arguments.

First, (α + β) - (β - α) = α + β - β + α = 2α. Then, (α + β) + (β - α) = α + β + β - α = 2β. Therefore, the product becomes [cos(2α) - cos(2β)] / 2. So the original expression simplifies to [cos(2α) - cos(2β)] / 2.

Hmm, that seems simpler. So instead of computing sin(α + β) and sin(β - α) separately, I can compute cos(2α) and cos(2β) and then take their difference. Let me check if this is correct. Let me verify with another approach.

Alternatively, if I compute sin(α + β) * sin(β - α), maybe I can also think of it as sin(α + β) * sin(β - α) = [sin β cos α + cos β sin α] * [sin β cos α - cos β sin α]. Wait, that's like (x + y)(x - y) where x = sin β cos α and y = cos β sin α. Therefore, this would be x² - y². So expanding that, it's (sin β cos α)^2 - (cos β sin α)^2.

Let me compute that. So sin² β cos² α - cos² β sin² α. Hmm, maybe I can factor this expression. Let me see. Alternatively, maybe I can use the identity for cos(2α) and cos(2β). Let me recall that cos(2θ) = 1 - 2 sin² θ, or 2 cos² θ - 1. Also, cos(2θ) = cos² θ - sin² θ. Wait, so if I take cos(2α) - cos(2β) = [cos² α - sin² α] - [cos² β - sin² β] = (cos² α - cos² β) - (sin² α - sin² β). Hmm, not sure if that helps.

But earlier, I had that the product sin(α + β) * sin(β - α) equals [cos(2α) - cos(2β)] / 2. Let me check if that's consistent with the other approach. Let me compute [cos(2α) - cos(2β)] / 2. If I use the identity for cos(2α) - cos(2β), which is -2 sin(α + β) sin(α - β). Wait, there's another identity: cos A - cos B = -2 sin[(A + B)/2] sin[(A - B)/2]. So cos(2α) - cos(2β) = -2 sin[(2α + 2β)/2] sin[(2α - 2β)/2] = -2 sin(α + β) sin(α - β). Therefore, [cos(2α) - cos(2β)] / 2 = - sin(α + β) sin(α - β). But the original expression is sin(α + β) * sin(β - α). Note that sin(β - α) = - sin(α - β). Therefore, sin(α + β) * sin(β - α) = - sin(α + β) sin(α - β). Which matches the previous result. Therefore, the original expression is equal to [cos(2α) - cos(2β)] / 2.

So either way, I can compute it as [cos(2α) - cos(2β)] / 2. Maybe this is easier? Let me try that.

So first, I need to compute cos(2α) and cos(2β). Since α = arcsin 0.4, then sin α = 0.4. Similarly, sin β = 0.5.

To find cos(2α), I can use the identity cos(2θ) = 1 - 2 sin² θ. Similarly, cos(2β) = 1 - 2 sin² β.

Alternatively, cos(2θ) = 2 cos² θ - 1. Either way, I need to find cos α and cos β.

Since α = arcsin 0.4, then cos α = sqrt(1 - sin² α) = sqrt(1 - 0.16) = sqrt(0.84). Similarly, cos β = sqrt(1 - sin² β) = sqrt(1 - 0.25) = sqrt(0.75).

But wait, arcsin returns values between -π/2 and π/2, so cos α and cos β are positive. Therefore, cos α = sqrt(1 - 0.16) = sqrt(0.84). Let me compute sqrt(0.84). Well, 0.84 is 84/100 = 21/25. Wait, sqrt(21)/5? Wait, sqrt(21)/5 squared is 21/25 = 0.84. Yes. So sqrt(0.84) is sqrt(21)/5. Similarly, sqrt(0.75) is sqrt(3)/2.

Therefore, cos α = sqrt(21)/5 and cos β = sqrt(3)/2.

So let me compute cos(2α) using the identity cos(2θ) = 1 - 2 sin² θ. For cos(2α): 1 - 2*(0.4)^2 = 1 - 2*(0.16) = 1 - 0.32 = 0.68. Alternatively, using cos(2α) = 2 cos² α - 1. Let me check that. 2*(sqrt(21)/5)^2 - 1 = 2*(21/25) - 1 = 42/25 - 1 = 17/25 = 0.68. Yes, same result.

Similarly, cos(2β) = 1 - 2*(0.5)^2 = 1 - 2*(0.25) = 1 - 0.5 = 0.5. Alternatively, 2*(sqrt(3)/2)^2 - 1 = 2*(3/4) - 1 = 3/2 - 1 = 1/2. Correct.

Therefore, cos(2α) - cos(2β) = 0.68 - 0.5 = 0.18. Then divide by 2: 0.18 / 2 = 0.09. Therefore, the original expression is 0.09.

Wait, but let me check if that's correct. Let me verify with another approach. Let me compute sin(α + β) and sin(β - α) separately and multiply them.

First, compute α = arcsin(0.4). Let me find sin(α + β). Using the formula sin(A + B) = sin A cos B + cos A sin B. So sin α = 0.4, cos α = sqrt(21)/5 ≈ sqrt(21)/5 ≈ 4.583/5 ≈ 0.9166. Wait, sqrt(21) is approx 4.58366, so divided by 5 is approx 0.9166. Similarly, sin β = 0.5, cos β = sqrt(3)/2 ≈ 0.8660.

Therefore, sin(α + β) = sin α cos β + cos α sin β = 0.4*(sqrt(3)/2) + (sqrt(21)/5)*0.5. Let me compute that.

First term: 0.4*(sqrt(3)/2) = 0.2*sqrt(3) ≈ 0.2*1.732 ≈ 0.3464.

Second term: (sqrt(21)/5)*0.5 = sqrt(21)/10 ≈ 4.583/10 ≈ 0.4583.

Adding them together: approx 0.3464 + 0.4583 ≈ 0.8047.

Now compute sin(β - α) = sin β cos α - cos β sin α.

Which is 0.5*(sqrt(21)/5) - (sqrt(3)/2)*0.4.

First term: 0.5*(sqrt(21)/5) = sqrt(21)/10 ≈ 0.4583.

Second term: (sqrt(3)/2)*0.4 = 0.4*(sqrt(3)/2) = 0.2*sqrt(3) ≈ 0.3464.

Therefore, sin(β - α) ≈ 0.4583 - 0.3464 ≈ 0.1119.

Then multiplying sin(α + β) * sin(β - α) ≈ 0.8047 * 0.1119 ≈ Let me compute that.

First, 0.8 * 0.1119 ≈ 0.0895, and 0.0047 * 0.1119 ≈ approx 0.000526. So total approx 0.0895 + 0.0005 ≈ 0.09. Which matches the previous result of 0.09.

So that seems to confirm that the answer is 0.09. But let me check if there's an exact value.

Earlier, we had [cos(2α) - cos(2β)] / 2 = (0.68 - 0.5)/2 = 0.18 / 2 = 0.09. But 0.68 is 17/25, and 0.5 is 1/2. Therefore, 17/25 - 1/2 = (34 - 25)/50 = 9/50. Then divided by 2 is 9/100, which is 0.09. So exact value is 9/100.

Therefore, the answer is 9/100, which is 0.09.

Alternatively, maybe I can check using exact expressions. Let me try that.

We had cos(2α) = 17/25 and cos(2β) = 1/2. Therefore, cos(2α) - cos(2β) = 17/25 - 1/2 = (34 - 25)/50 = 9/50. Then divided by 2 is 9/100. So yes, exactly 9/100.

Alternatively, if I compute sin(α + β) * sin(β - α) using exact terms.

Earlier, we had sin(α + β) = 0.4*(sqrt(3)/2) + (sqrt(21)/5)*0.5. Let me write that as fractions.

0.4 is 2/5, so 2/5 * sqrt(3)/2 = (2/5)*(sqrt(3)/2) = sqrt(3)/5.

Similarly, sqrt(21)/5 * 0.5 is sqrt(21)/5 * 1/2 = sqrt(21)/10.

Therefore, sin(α + β) = sqrt(3)/5 + sqrt(21)/10.

Similarly, sin(β - α) = sqrt(21)/10 - sqrt(3)/5.

Therefore, multiplying them: [sqrt(3)/5 + sqrt(21)/10] * [sqrt(21)/10 - sqrt(3)/5].

This is of the form (a + b)(b - a) = b² - a², where a = sqrt(3)/5 and b = sqrt(21)/10.

Wait, let me check. Let me denote a = sqrt(3)/5 and b = sqrt(21)/10. Then the expression is (a + b)(b - a) = b² - a².

Compute b²: (sqrt(21)/10)^2 = 21/100.

Compute a²: (sqrt(3)/5)^2 = 3/25 = 12/100.

Therefore, b² - a² = 21/100 - 12/100 = 9/100.

Exactly! Therefore, the product is 9/100.

So that's the exact value. Therefore, the answer is 9/100.

Therefore, regardless of the method, I get 9/100.

Alternatively, maybe there's another way to think about this problem. Let me think.

Suppose I let x = arcsin 0.4 and y = arcsin 0.5. Then the problem is sin(x + y) * sin(y - x).

Alternatively, note that sin(x + y) * sin(y - x) = [sin y cos x + cos y sin x][sin y cos x - cos y sin x] = (sin y cos x)^2 - (cos y sin x)^2.

Which is sin² y cos² x - cos² y sin² x.

Factor this expression. Let me see.

sin² y cos² x - cos² y sin² x = sin² y cos² x - cos² y sin² x.

Hmm, maybe factor sin² y cos² x - cos² y sin² x. Let me factor sin² y cos² x - cos² y sin² x.

Alternatively, factor out sin² y cos² x - cos² y sin² x = sin² y cos² x - cos² y sin² x.

Let me write this as sin² y cos² x - cos² y sin² x = sin² y cos² x - sin² x cos² y.

Which is equal to sin² y cos² x - sin² x cos² y.

This can be written as (sin y cos x - sin x cos y)(sin y cos x + sin x cos y). Wait, but that's the same as sin(y - x) * sin(y + x). Which brings us back to the original expression. So that's circular.

Alternatively, maybe express in terms of sin(y - x) and sin(y + x). But we already did that.

Alternatively, maybe use the identity sin² A - sin² B = sin(A - B) sin(A + B). Wait, yes! Because sin² A - sin² B = (sin A - sin B)(sin A + sin B). But also, using the identity sin² A - sin² B = sin(A - B) sin(A + B). Let me check that.

Yes, because sin(A - B) sin(A + B) = [sin A cos B - cos A sin B][sin A cos B + cos A sin B] = sin² A cos² B - cos² A sin² B. Which is not exactly the same as sin² A - sin² B. Wait, unless we manipulate further.

Alternatively, let me recall that sin² A - sin² B = (sin A - sin B)(sin A + sin B). But also, using the identity sin² A - sin² B = (sin A - sin B)(sin A + sin B) = [2 cos((A + B)/2) sin((A - B)/2)][2 sin((A + B)/2) cos((A - B)/2)] = 4 sin((A + B)/2) cos((A + B)/2) sin((A - B)/2) cos((A - B)/2) = sin(A + B) sin(A - B).

Yes! Therefore, sin² A - sin² B = sin(A + B) sin(A - B). Therefore, in our case, sin² y cos² x - cos² y sin² x = sin² y cos² x - sin² x cos² y = [sin y cos x]^2 - [sin x cos y]^2 = sin² y cos² x - sin² x cos² y = sin(y + x) sin(y - x). Wait, but that's the original expression. Wait, no. Wait, according to the identity, sin² A - sin² B = sin(A + B) sin(A - B). But here, we have [sin y cos x]^2 - [sin x cos y]^2. Let me think of this as sin² y cos² x - sin² x cos² y. Let me factor this as cos² x sin² y - cos² y sin² x.

Let me factor out sin² y cos² x - sin² x cos² y. Let me write this as sin² y cos² x - sin² x cos² y = sin² y cos² x - sin² x cos² y.

Let me factor this expression. Let me think of it as (sin y cos x - sin x cos y)(sin y cos x + sin x cos y). Which is sin(y - x) * sin(y + x). Which is the original expression. Therefore, again, circular.

But earlier, we found that this expression equals 9/100.

Alternatively, maybe I can compute sin(α + β) and sin(β - α) numerically with more precision. Let me check.

First, compute α = arcsin(0.4). Let me compute α in radians.

Well, arcsin(0.4) is approximately... Let me recall that sin(0.4115 radians) ≈ 0.4. Because sin(0.4115) ≈ 0.4. Let me check with calculator.

But since I don't have a calculator here, but maybe I can use known values. For example, sin(π/6) = 0.5, which is 0.5. So 0.4 is less than that. π/6 is approx 0.5236 radians. So arcsin(0.4) is less than π/6. Let me approximate.

Alternatively, maybe use Taylor series? Not sure. Alternatively, maybe just accept that we can compute the exact value as 9/100.

But since we already have two methods giving 9/100, and the approximate decimal calculation also gives 0.09, which is 9/100, I think that's solid.

Therefore, the answer is 9/100.

But just to make sure, let me check with another approach. Let me compute the numerical values of α and β, then compute the angles, then compute the sines.

First, compute α = arcsin(0.4). Let me use a calculator (pretending I have one). Let me recall that arcsin(0.4) ≈ 0.4115 radians. Similarly, arcsin(0.5) is π/6 ≈ 0.5236 radians.

Therefore, α + β ≈ 0.4115 + 0.5236 ≈ 0.9351 radians. Then sin(0.9351) ≈ sin(0.9351). Let me recall that sin(π/3) ≈ 0.8660, π/3 ≈ 1.0472 radians. So 0.9351 is less than π/3. Let me approximate sin(0.9351).

Alternatively, use the value we calculated earlier: approx 0.8047. Let me check with calculator steps.

Alternatively, use the exact expression: sin(α + β) = sqrt(3)/5 + sqrt(21)/10 ≈ 1.732/5 + 4.583/10 ≈ 0.3464 + 0.4583 ≈ 0.8047.

Similarly, sin(β - α) ≈ 0.1119.

Multiplying them gives approx 0.8047 * 0.1119 ≈ 0.09.

Alternatively, if I compute 0.8047 * 0.1119:

First, 0.8 * 0.1119 = 0.08952

Then 0.0047 * 0.1119 ≈ 0.000526

Total ≈ 0.08952 + 0.000526 ≈ 0.090046, which is approx 0.09.

Therefore, 0.09 is accurate.

Therefore, the exact value is 9/100, which is 0.09.

Therefore, the answer is 9/100.

But let me check once more with another identity.

We had earlier that the product sin(α + β) * sin(β - α) equals [cos(2α) - cos(2β)] / 2.

We found cos(2α) = 17/25 and cos(2β) = 1/2. Therefore, [17/25 - 1/2]/2 = [ (34 - 25)/50 ] / 2 = (9/50)/2 = 9/100.

Yes, that's correct.

Alternatively, if I use the formula for sin(A)sin(B) = [cos(A - B) - cos(A + B)] / 2.

Wait, in our case, A = α + β and B = β - α. Therefore, sin(A)sin(B) = [cos(A - B) - cos(A + B)] / 2.

Compute A - B = (α + β) - (β - α) = 2α.

A + B = (α + β) + (β - α) = 2β.

Therefore, sin(A)sin(B) = [cos(2α) - cos(2β)] / 2. Which is exactly what we had before.

Therefore, this confirms again that the value is 9/100.

Therefore, I think I can be confident that the answer is 9/100.

But just to make sure, let me check if there's any possible mistake in the calculation of cos(2α) and cos(2β).

For α = arcsin(0.4), sin α = 0.4. Therefore, cos α = sqrt(1 - 0.16) = sqrt(0.84). Then cos(2α) = 1 - 2*(0.4)^2 = 1 - 0.32 = 0.68. Which is 17/25. Because 0.68 = 68/100 = 17/25. Correct.

Similarly, for β = arcsin(0.5), sin β = 0.5, so cos β = sqrt(1 - 0.25) = sqrt(0.75) = sqrt(3)/2. Then cos(2β) = 1 - 2*(0.5)^2 = 1 - 0.5 = 0.5 = 1/2. Correct.

Therefore, cos(2α) - cos(2β) = 17/25 - 1/2 = 34/50 - 25/50 = 9/50. Divided by 2 is 9/100.

Yes, that's correct.

Therefore, the answer is 9/100.

Alternatively, if I convert 9/100 to decimal, it's 0.09, which matches the approximate calculation.

Therefore, I think this is solid.

So, the final answer is 9/100, which can be written as \boxed{\dfrac{9}{100}}.

**Final Answer**
\boxed{\dfrac{9}{100}}

To evaluate the expression:

$$
\sin (\arcsin 0.4 + \arcsin 0.5) \cdot \sin (\arcsin 0.5 - \arcsin 0.4),
$$

we begin by introducing substitutions to simplify the notation. Let:

- $ \alpha = \arcsin 0.4 $, so $ \sin \alpha = 0.4 $,
- $ \beta = \arcsin 0.5 $, so $ \sin \beta = 0.5 $.

Then the expression becomes:

$$
\sin(\alpha + \beta) \cdot \sin(\beta - \alpha).
$$

---

### Step 1: Use Trigonometric Identities

We apply the **product-to-sum identity** for sine:

$$
\sin A \sin B = \frac{1}{2} [\cos(A - B) - \cos(A + B)].
$$

Let $ A = \alpha + \beta $ and $ B = \beta - \alpha $. Then:

$$
\sin(\alpha + \beta) \cdot \sin(\beta - \alpha) = \frac{1}{2} [\cos((\alpha + \beta) - (\beta - \alpha)) - \cos((\alpha + \beta) + (\beta - \alpha))].
$$

Simplify the arguments:

- $ (\alpha + \beta) - (\beta - \alpha) = 2\alpha $,
- $ (\alpha + \beta) + (\beta - \alpha) = 2\beta $.

So the expression becomes:

$$
\frac{1}{2} [\cos(2\alpha) - \cos(2\beta)].
$$

---

### Step 2: Compute $ \cos(2\alpha) $ and $ \cos(2\beta) $

We use the identity:

$$
\cos(2\theta) = 1 - 2\sin^2 \theta.
$$

- For $ \alpha $, $ \sin \alpha = 0.4 $, so:
  $$
  \cos(2\alpha) = 1 - 2(0.4)^2 = 1 - 2(0.16) = 1 - 0.32 = 0.68.
  $$

- For $ \beta $, $ \sin \beta = 0.5 $, so:
  $$
  \cos(2\beta) = 1 - 2(0.5)^2 = 1 - 2(0.25) = 1 - 0.5 = 0.5.
  $$

---

### Step 3: Plug into the Expression

Now compute:

$$
\frac{1}{2} [\cos(2\alpha) - \cos(2\beta)] = \frac{1}{2} [0.68 - 0.5] = \frac{1}{2} \cdot 0.18 = 0.09.
$$

---

### Step 4: Express as a Fraction

Since $ 0.09 = \frac{9}{100} $, the exact value of the expression is:

$$
\boxed{\dfrac{9}{100}}
$$

Solution:
From the angle addition and subtraction formulas,
\begin{align*}
\sin (x + y) &= \sin x \cos y + \cos x \sin y, \\
\sin (x - y) &= \sin x \cos y - \cos x \sin y,
\end{align*}so
\begin{align*}
\sin (x + y) \sin (x - y) &= (\sin x \cos y + \cos x \sin y)(\sin x \cos y - \cos x \sin y) \\
&= \sin^2 x \cos^2 y + \sin x \cos x \sin y \cos y - \sin x \cos x \sin y \cos y - \cos^2 x \sin^2 y \\
&= \sin^2 x (1 - \sin^2 y) - (1 - \sin^2 x) \sin^2 y \\
&= \sin^2 x - \sin^2 x \sin^2 y - \sin^2 y + \sin^2 x \sin^2 y \\
&= \sin^2 x - \sin^2 y.
\end{align*}Taking $x = \arcsin 0.5$ and $y = \arcsin 0.4,$ we get
\begin{align*}
\sin (\arcsin 0.5 + \arcsin 0.4) \cdot \sin (\arcsin 0.5 - \arcsin 0.4) &= \sin^2 (\arcsin 0.5) - \sin^2 (\arcsin 0.4) \\
&= 0.5^2 - 0.4^2 \\
&= 0.09 = \boxed{\frac{9}{100}}.
\end{align*}


Explanation:
Okay, so I need to figure out how many different license plates can be made with the given conditions. Let me start by understanding the problem again. The license plate has 2 different letters followed by 2 different one-digit numbers. The letters are from A through Z, which I know there are 26 letters in the English alphabet. The numbers are from 0 to 9, so that's 10 different digits.

First, let me break down the problem into parts. The license plate has two sections: letters and numbers. The letters part has two different letters, and the numbers part has two different digits. So I need to calculate the number of possibilities for the letters and then the number for the digits, and then multiply them together because for each combination of letters, there are multiple combinations of numbers, and vice versa.

Starting with the letters. The first thing I need to consider is that the two letters must be different. So, for the first letter, there are 26 possible choices since there are 26 letters. Now, for the second letter, since it has to be different from the first one, there are 25 remaining choices. So, does that mean the number of possible combinations for the letters is 26 multiplied by 25? Let me think.

Yes, because for each of the 26 choices of the first letter, there are 25 choices for the second letter. So that would be 26 * 25. But wait, is the order important here? Like, is "AB" different from "BA"? The problem says "license plate has 2 different letters", so I think the order matters because license plates are ordered. For example, "AB" is different from "BA". Therefore, permutations are involved here, not combinations.

So, if I think in terms of permutations, the number of ways to arrange 2 different letters out of 26 is P(26, 2) which is calculated as 26 * 25. That's the same as what I thought earlier. So that part seems straightforward.

Now moving on to the numbers. The problem says two different one-digit numbers. So similar to the letters, the first digit can be any of the 10 digits (0-9), and the second digit has to be different from the first one. Therefore, for the first digit, there are 10 choices, and for the second digit, there are 9 choices left. So, the number of combinations for the digits would be 10 * 9. Again, since the order matters here as well. For example, "12" is different from "21", so permutations again.

Therefore, the total number of license plates would be the number of letter combinations multiplied by the number of digit combinations. So that would be (26 * 25) * (10 * 9). Let me compute that.

First, let me calculate 26 * 25. 26 times 25. Well, 25 times 20 is 500, and 25 times 6 is 150, so 500 + 150 = 650. So 26 * 25 = 650.

Then, 10 * 9 is 90.

So multiplying those two results together: 650 * 90. Let me compute that.

Breaking it down: 650 * 90. Well, 650 * 9 = 5850, so 650 * 90 is 5850 * 10 = 58,500.

Wait, let me check that again. 650 * 90. Alternatively, 650 * 9 * 10. 650 * 9: 600*9=5400, 50*9=450, so 5400 + 450 = 5850. Then times 10 is 58,500. Yes, that seems right.

So, according to this calculation, there are 58,500 possible license plates.

But let me just make sure I didn't make any mistakes in my reasoning.

First, letters: 2 different letters. Since they are different and order matters, permutations. So P(26, 2) = 26*25 = 650. That seems correct.

Numbers: 2 different digits. Similarly, permutations. P(10, 2) = 10*9 = 90. Correct.

Total combinations: 650 * 90 = 58,500.

Alternatively, maybe I can think of it as:

For the letters: first letter 26 choices, second letter 25 choices. For numbers: first digit 10 choices, second digit 9 choices. So total is 26 * 25 * 10 * 9. Which is the same as (26*25)*(10*9) = 650 * 90 = 58,500.

Alternatively, maybe I can compute 26*25*10*9 step by step. Let me try that.

26 * 25 = 650. Then 650 * 10 = 6,500. Then 6,500 * 9. Let me compute 6,500 * 9. 6,000 * 9 = 54,000. 500 * 9 = 4,500. So total is 54,000 + 4,500 = 58,500. Same result.

So that seems consistent.

But let me check if there's another way to think about this. For example, sometimes when people count permutations, they might confuse combinations with permutations. Let me make sure that the problem is indeed about permutations.

The problem says "2 different letters" followed by "2 different one-digit numbers". So, for the letters, since they are different and the order matters (as license plates are ordered), it's permutations. Similarly for the numbers.

Alternatively, if the problem had said "2 letters, not necessarily different", then it would be 26*26 for letters, and similarly for numbers. But since they are different, we have to subtract the cases where they are the same.

But in this case, since they are different, the calculation is permutations.

Another way to think about the letters: the number of ways to choose 2 different letters where order matters is 26*25. Similarly for numbers.

Alternatively, if we thought of it as combinations first, then multiplying by the number of arrangements. For example, the number of combinations of letters is C(26, 2) = (26*25)/2. Then, since each combination can be arranged in 2 ways, so total permutations would be C(26, 2)*2! = (26*25)/2 * 2 = 26*25. Which is the same as before. So that's another way to get the same result.

Same with numbers: combinations C(10, 2) = (10*9)/2, multiplied by 2! gives 10*9.

So that's another way to think about it, but arrives at the same answer.

Therefore, I think my calculation is correct.

But let me check if there's any possibility that the numbers can have leading zeros? For example, is "01" considered a valid two-digit number? Well, the problem says "two different one-digit numbers". So each is a single digit, so they can be 0-9. So, for example, the first number could be 0, and the second could be 1, so "01" would be allowed. So there's no restriction on leading zeros. Therefore, the calculation for numbers is correct.

Similarly, for letters, there's no restriction on letters, so all letters from A-Z are allowed.

Therefore, I think my calculation is correct.

So, the total number of license plates is 26 * 25 * 10 * 9 = 58,500.

Wait, just to make sure, let me check with smaller numbers. Let's say instead of 26 letters and 10 digits, suppose we have 3 letters and 2 digits. How many license plates would there be?

Using the same logic: letters: 3*2=6, digits: 2*1=2, total 6*2=12.

Let me list them. Letters: AB, BA, AC, CA, BC, CB. Digits: 01, 10, 02, 20, 03, 30, 12, 21, 13, 31, 23, 32. Wait, hold on, if digits are 0-2 (for example), then two different digits would be 01, 10, 02, 20, 12, 21. So 6 possibilities. So total license plates would be 6 letters * 6 digits = 36. Wait, but according to my previous calculation, with 3 letters and 2 digits, letters would be 3*2=6, digits would be 2*1=2? Wait, no, if digits are 0-2, that's 3 digits. So for two different digits, it's P(3,2)=3*2=6. So total license plates would be 6*6=36. Which is correct.

But if I had said digits are 0-9, then with two different digits, it's 10*9=90. So that seems correct.

So in my original problem, with 26 letters and 10 digits, the calculation is 26*25*10*9=58,500.

Therefore, I think that is correct.

Alternatively, if someone thought that the numbers are two-digit numbers, meaning from 00 to 99, but with different digits. But the problem says "2 different one-digit numbers", so it's two separate digits, not a two-digit number. So for example, if it was a two-digit number, it would be from 00 to 99, but with different digits, so 90 possibilities (as 10*9). But since it's two different one-digit numbers, it's the same thing. So that's consistent.

Therefore, I think my answer is correct.

So, the final answer is 58,500.

**Final Answer**
The number of possible license plates is \boxed{58500}.

To determine how many different license plates can be formed, we analyze the structure of the license plate: it consists of **2 different letters** followed by **2 different digits**. Each part must be considered separately, and the total number of combinations is the product of the number of valid combinations for each part.

---

### **Step 1: Counting the Letter Combinations**

We are given that the license plate has **2 different letters** from the English alphabet (A through Z), which contains **26 letters**.

- The **first letter** can be any of the 26 letters.
- The **second letter** must be **different** from the first, so it has **25 choices**.

Since the **order matters** (e.g., "AB" is different from "BA"), we are dealing with **permutations** of 2 letters from 26.

$$
\text{Number of letter combinations} = P(26, 2) = 26 \times 25 = 650
$$

---

### **Step 2: Counting the Digit Combinations**

Next, we consider the **2 different digits** from the set {0, 1, ..., 9}, which contains **10 digits**.

- The **first digit** can be any of the 10 digits.
- The **second digit** must be **different** from the first, so it has **9 choices**.

Again, since the **order matters** (e.g., "12" is different from "21"), we are dealing with **permutations** of 2 digits from 10.

$$
\text{Number of digit combinations} = P(10, 2) = 10 \times 9 = 90
$$

---

### **Step 3: Total Number of License Plates**

To find the total number of possible license plates, we multiply the number of letter combinations by the number of digit combinations:

$$
\text{Total license plates} = 650 \times 90 = 58,500
$$

---

### **Final Answer**

$$
\boxed{58500}
$$

[Assistant]
Explanation:
Okay, so I need to figure out how many different license plates can be made with the given conditions. Let me start by understanding the problem again. The license plate has 2 different letters followed by 2 different one-digit numbers. The letters are from A through Z, which I know there are 26 letters in the English alphabet. The numbers are from 0 to 9, so that's 10 different digits.

First, let me break down the problem into parts. The license plate has two sections: letters and numbers. The letters part has two different letters, and the numbers part has two different digits. So I need to calculate the number of possibilities for the letters and then the number for the digits, and then multiply them together because for each combination of letters, there are multiple combinations of numbers, and vice versa.

Starting with the letters. The first thing I need to consider is that the two letters must be different. So, for the first letter, there are 26 possible choices since there are 26 letters. Now, for the second letter, since it has to be different from the first one, there are 25 remaining choices. So, does that mean the number of possible combinations for the letters is 26 multiplied by 25? Let me think.

Yes, because for each of the 26 choices of the first letter, there are 25 choices for the second letter. So that would be 26 * 25. But wait, is the order important here? Like, is "AB" different from "BA"? The problem says "license plate has 2 different letters", so I think the order matters because license plates are ordered. For example, "AB" is different from "BA". Therefore, permutations are involved here, not combinations.

So, if I think in terms of permutations, the number of ways to arrange 2 different letters out of 26 is P(26, 2) which is calculated as 26 * 25. That's the same as what I thought earlier. So that part seems straightforward.

Now moving on to the numbers. The problem says two different one-digit numbers. So similar to the letters, the first digit can be any of the 10 digits (0-9), and the second digit has to be different from the first one. Therefore, for the first digit, there are 10 choices, and for the second digit, there are 9 choices left. So, the number of combinations for the digits would be 10 * 9. Again, since the order matters here as well. For example, "12" is different from "21", so permutations again.

Therefore, the total number of license plates would be the number of letter combinations multiplied by the number of digit combinations. So that would be (26 * 25) * (10 * 9). Let me compute that.

First, let me calculate 26 * 25. 26 times 25. Well, 25 times 20 is 500, and 25 times 6 is 150, so 500 + 150 = 650. So 26 * 25 = 650.

Then, 10 * 9 is 90.

So multiplying those two results together: 650 * 90. Let me compute that.

Breaking it down: 650 * 90. Well, 650 * 9 = 5850, so 650 * 90 is 5850 * 10 = 58,500.

Wait, let me check that again. 650 * 90. Alternatively, 650 * 9 * 10. 650 * 9: 600*9=5400, 50*9=450, so 5400 + 450 = 5850. Then times 10 is 58,500. Yes, that seems right.

So, according to this calculation, there are 58,500 possible license plates.

But let me just make sure I didn't make any mistakes in my reasoning.

First, letters: 2 different letters. Since they are different and order matters, permutations. So P(26, 2) = 26*25 = 650. That seems correct.

Numbers: 2 different digits. Similarly, permutations. P(10, 2) = 10*9 = 90. Correct.

Total combinations: 650 * 90 = 58,500.

Alternatively, maybe I can think of it as:

For the letters: first letter 26 choices, second letter 25 choices. For numbers: first digit 10 choices, second digit 9 choices. So total is 26 * 25 * 10 * 9. Which is the same as (26*25)*(10*9) = 650 * 90 = 58,500.

Alternatively, maybe I can compute 26*25*10*9 step by step. Let me try that.

26 * 25 = 650. Then 650 * 10 = 6,500. Then 6,500 * 9. Let me compute 6,500 * 9. 6,000 * 9 = 54,000. 500 * 9 = 4,500. So total is 54,000 + 4,500 = 58,500. Same result.

So that seems consistent.

But let me check if there's another way to think about this. For example, sometimes when people count permutations, they might confuse combinations with permutations. Let me make sure that the problem is indeed about permutations.

The problem says "2 different letters" followed by "2 different one-digit numbers". So, for the letters, since they are different and the order matters (as license plates are ordered), it's permutations. Similarly for the numbers.

Alternatively, if the problem had said "2 letters, not necessarily different", then it would be 26*26 for letters, and similarly for numbers. But since they are different, we have to subtract the cases where they are the same.

But in this case, since they are different, the calculation is permutations.

Another way to think about the letters: the number of ways to choose 2 different letters where order matters is 26*25. Similarly for numbers.

Alternatively, if we thought of it as combinations first, then multiplying by the number of arrangements. For example, the number of combinations of letters is C(26, 2) = (26*25)/2. Then, since each combination can be arranged in 2 ways, so total permutations would be C(26, 2)*2! = (26*25)/2 * 2 = 26*25. Which is the same as before. So that's another way to get the same result.

Same with numbers: combinations C(10, 2) = (10*9)/2, multiplied by 2! gives 10*9.

So that's another way to think about it, but arrives at the same answer.

Therefore, I think my calculation is correct.

But let me check if there's any possibility that the numbers can have leading zeros? For example, is "01" considered a valid two-digit number? Well, the problem says "two different one-digit numbers". So each is a single digit, so they can be 0-9. So, for example, the first number could be 0, and the second could be 1, so "01" would be allowed. So there's no restriction on leading zeros. Therefore, the calculation for numbers is correct.

Similarly, for letters, there's no restriction on letters, so all letters from A-Z are allowed.

Therefore, I think my calculation is correct.

So, the total number of license plates is 26 * 25 * 10 * 9 = 58,500.

Wait, just to make sure, let me check with smaller numbers. Let's say instead of 26 letters and 10 digits, suppose we have 3 letters and 2 digits. How many license plates would there be?

Using the same logic: letters: 3*2=6, digits: 2*1=2, total 6*2=12.

Let me list them. Letters: AB, BA, AC, CA, BC, CB. Digits: 01, 10, 02, 20, 03, 30, 12, 21, 13, 31, 23, 32. Wait, hold on, if digits are 0-2 (for example), then two different digits would be 01, 10, 02, 20, 12, 21. So 6 possibilities. So total license plates would be 6 letters * 6 digits = 36. Wait, but according to my previous calculation, with 3 letters and 2 digits, letters would be 3*2=6, digits would be 2*1=2? Wait, no, if digits are 0-2, that's 3 digits. So for two different digits, it's P(3,2)=3*2=6. So total license plates would be 6*6=36. Which is correct.

But if I had said digits are 0-9, then with two different digits, it's 10*9=90. So that seems correct.

So in my original problem, with 26 letters and 10 digits, the calculation is 26*25*10*9=58,500.

Therefore, I think that is correct.

Alternatively, if someone thought that the numbers are two-digit numbers, meaning from 00 to 99, but with different digits. But the problem says "2 different one-digit numbers", so it's two separate digits, not a two-digit number. So for example, if it was a two-digit number, it would be from 00 to 99, but with different digits, so 90 possibilities (as 10*9). But since it's two different one-digit numbers, it's the same thing. So that's consistent.

Therefore, I think my answer is correct.

So, the final answer is 58,500.

**Final Answer**
The number of possible license plates is \boxed{58500}.

To determine how many different license plates can be formed, we analyze the structure of the license plate: it consists of **2 different letters** followed by **2 different digits**. Each part must be considered separately, and the total number of combinations is the product of the number of valid combinations for each part.

---

### **Step 1: Counting the Letter Combinations**

We are given that the license plate has **2 different letters** from the English alphabet (A through Z), which contains **26 letters**.

- The **first letter** can be any of the 26 letters.
- The **second letter** must be **different** from the first, so it has **25 choices**.

Since the **order matters** (e.g., "AB" is different from "BA"), we are dealing with **permutations** of 2 letters from 26.

$$
\text{Number of letter combinations} = P(26, 2) = 26 \times 25 = 650
$$

---

### **Step 2: Counting the Digit Combinations**

Next, we consider the **2 different digits** from the set {0, 1, ..., 9}, which contains **10 digits**.

- The **first digit** can be any of the 10 digits.
- The **second digit** must be **different** from the first, so it has **9 choices**.

Again, since the **order matters** (e.g., "12" is different from "21"), we are dealing with **permutations** of 2 digits from 10.

$$
\text{Number of digit combinations} = P(10, 2) = 10 \times 9 = 90
$$

---

### **Step 3: Total Number of License Plates**

To find the total number of possible license plates, we multiply the number of letter combinations by the number of digit combinations:

$$
\text{Total license plates} = 650 \times 90 = 58,500
$$

---

### **Final Answer**

$$
\boxed{58500}
$$

Solution: The first letter can be any of the 26 letters of the alphabet, while the second letter can be any of the 25 remaining letters. The first digit can be any of the 10 digits, while the second digit can be any of the 9 remaining digits. The number of license plates is $26\times 25\times 10\times 9=\boxed{58,500}$.