Verification of (x^3)(y^3) = (x*y)(x^2 - xy + y^2)

To verify the given equation (x^3)(y^3) = (x*y)(x^2 - xy + y^2), we need to simplify both sides of the equation and check if they are equal.

Expanding the left-hand side (LHS):
(x^3)(y^3) can be expanded using the laws of exponents. When we multiply two numbers with the same base, we add their exponents.

(x^3)(y^3) = (x * x * x)(y * y * y)
= x^3 * y^3

Expanding the right-hand side (RHS):
(x*y)(x^2 - xy + y^2) can be expanded using the distributive property of multiplication over addition.

(x*y)(x^2 - xy + y^2) = (x*y*x^2) + (x*y*(-xy)) + (x*y*y^2)
= x^3*y + (-x^2 * y^2) + (xy^3)
= x^3*y - x^2*y^2 + xy^3

Now, we can see that the LHS is equal to x^3 * y^3, and the RHS is equal to x^3*y - x^2*y^2 + xy^3.

Comparing LHS and RHS:
x^3*y^3 = x^3*y - x^2*y^2 + xy^3

By comparing the LHS and RHS, we can see that they are equal.

Therefore, the equation (x^3)(y^3) = (x*y)(x^2 - xy + y^2) is verified.

Summary:
- We verified the given equation (x^3)(y^3) = (x*y)(x^2 - xy + y^2).
- By expanding and simplifying both sides of the equation, we found that LHS = x^3*y^3 and RHS = x^3*y - x^2*y^2 + xy^3.
- By comparing LHS and RHS, we concluded that they are equal, thus verifying the equation.