https://file.aiquickdraw.com/mathbot/file/1726607986706mgfqli2r.jpg

The question in the image is:

**If \(x\) and \(y\) are real numbers such that \(x^2 + y^2 = 13\) and \(xy = 3\), what is the value of \(x^4 + y^4\)?**

### Solution:

We are tasked with finding \(x^4 + y^4\). First, notice that:

\[
x^4 + y^4 = (x^2 + y^2)^2 - 2(xy)^2
\]

This formula comes from the identity:

\[
(x^2 + y^2)^2 = x^4 + y^4 + 2x^2y^2
\]

So, rearranging gives:

\[
x^4 + y^4 = (x^2 + y^2)^2 - 2x^2y^2
\]

We are given that:
- \(x^2 + y^2 = 13\)
- \(xy = 3\)

Now, let's substitute these values into the formula:

\[
x^4 + y^4 = (13)^2 - 2(3)^2
\]

First, calculate each term:

\[
13^2 = 169 \quad \text{and} \quad 2(3)^2 = 2 \times 9 = 18
\]

Now, substitute these back into the equation:

\[
x^4 + y^4 = 169 - 18 = 151
\]

Thus, the value of \(x^4 + y^4\) is:

\[
\boxed{151}
\]

Would you like further clarification, or do you have any additional questions?

### Follow-up Questions:
1. How would the result change if \(xy = 4\) instead of 3?
2. Can you derive a similar formula for \(x^6 + y^6\)?
3. What happens if the given condition is \(x^2 + y^2 = 25\) and \(xy = 7\)?
4. Can you generalize the formula for \(x^n + y^n\) for any even \(n\)?
5. What would \(x^4 + y^4\) be if both \(x\) and \(y\) were complex numbers?

### Tip:
When solving polynomial identities, always check if there's a way to express higher powers in terms of sums and products of the lower powers using algebraic identities!

If x and y are real numbers such that x^2 + y^2 = 13 and xy = 3, what is the value of x^4 + y^4?

Solve the algebraic identity problem where x^2 + y^2 = 13 and xy = 3 to find x^4 + y^4. This detailed solution explores key polynomial identities and algebraic concepts suitable for Grades 9-11.

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