Question

A^3+b^3=a+b=3 a*b=?​

Answer 1

Решение задания приложено

Answer 2

Let's solve the given equations step by step.

Equation 1: A^3 + B^3 = A + B = 3

Equation 2: A * B = ?

To find the value of A * B, we need to use the given equations and solve for A and B individually.

From the equation A^3 + B^3 = A + B = 3, we can rewrite it as:

(A + B)(A^2 - AB + B^2) = A + B

Since A + B = 3, we have:

3(A^2 - AB + B^2) = 3

Dividing both sides of the equation by 3 gives us:

A^2 - AB + B^2 = 1

Now, let's manipulate the equation further.

A^2 - AB + B^2 = 1

Multiplying both sides by (A - B) gives:

(A - B)(A^2 - AB + B^2) = (A - B)

Expanding the left side of the equation:

A^3 - A^2B + AB^2 - A^2B + AB^2 - B^3 = A - B

Simplifying:

A^3 - 2A^2B + 2AB^2 - B^3 = A - B

Now, let's use the other given equation A + B = 3.

We can rewrite it as:

(A + B)(A^2 - AB + B^2) = 3

Substituting A + B = 3, we have:

3(A^2 - AB + B^2) = 3

Dividing both sides by 3:

A^2 - AB + B^2 = 1

We can see that this equation is the same as the one we derived earlier. Therefore, we can substitute A^3 - 2A^2B + 2AB^2 - B^3 = A - B with 1.

A^3 - 2A^2B + 2AB^2 - B^3 = A - B (Equation 1) A^3 + B^3 = A + B = 3 (Equation 2)

Substituting Equation 2 into Equation 1:

3 - 2A^2B + 2AB^2 = 3

Simplifying:

-2A^2B + 2AB^2 = 0

Factoring out 2AB:

2AB(-A + B) = 0

Since 2AB cannot be zero (as it would imply A or B is zero), we have:

-A + B = 0

Rearranging the equation:

B = A

Now, we can substitute B = A into the equation A + B = 3:

A + A = 3

2A = 3

A = 3/2

Since B = A, we have B = 3/2 as well.

Finally, we can calculate A * B:

A * B = (3/2) * (3/2) = 9/4

Therefore, A * B is equal to 9/4.