Given the problem where the sum of $x$ and $y$ is 8, and the product of $x$ and $y$ is 2, we are asked to find $x^3 + y^3$.

Step 1: Use the identities

We can use the identity for the sum of cubes: $x^3 + y^3 = (x + y)(x^2 - xy + y^2)$ We know $x + y = 8$ and $xy = 2$.

Step 2: Find $x^2 + y^2$

We can use the identity: $x^2 + y^2 = (x + y)^2 - 2xy$ Substitute the given values: $x^2 + y^2 = 8^2 - 2(2) = 64 - 4 = 60$

Step 3: Substitute into the sum of cubes formula

Now substitute into the formula for $x^3 + y^3$: $x^3 + y^3 = (x + y)\left(x^2 - xy + y^2\right)$ $x^3 + y^3 = 8(60 - 2) = 8(58) = 464$

So, $x^3 + y^3 = 464$.

Would you like any more details or have any questions? Here are 5 related questions to consider:

1. How would the solution change if the sum of $x$ and $y$ were different?
2. What are the values of $x$ and $y$ that satisfy these conditions?
3. How would you approach this problem if only given the sum or the product of $x$ and $y$?
4. What is the value of $x^4 + y^4$ using the same approach?
5. How can the discriminant of the quadratic equation derived from these values help find $x$ and $y$?

Tip: Remember that identities like $x^2 + y^2 = (x + y)^2 - 2xy$ can simplify calculations involving sums and products of roots.