If the sum of x and y is 8 and the product of x and y is 2 what is x cubed plus y cubed

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.

Learn how to solve for x^3 + y^3 when given the sum of x and y as 8 and their product as 2. This problem involves using algebraic identities and formulas such as the sum of cubes and difference of squares. Suitable for advanced high school students.

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