https://file.aiquickdraw.com/mathbot/file/172599143714594dqt31f.jpg

The image contains several mathematical problems, each with different equations and instructions. Let's go through them one by one:

### Problem 1:
\[
a - \frac{1}{a} = 2
\]
You are tasked with solving for \(a\).

### Problem 2:
\[
x^2 + \frac{1}{x^2} = 3
\]
You need to find a solution for \(x\).

### Problem 3:
\[
x + y = 5 \quad \text{and} \quad xy = 6
\]
You are instructed to compute \(x^3 - y^3\).

### Problem 4:
Prove that:
\[
\frac{a^8 + 1}{a^4} = 34
\]
The goal is to show that this equation holds true.

### Problem 5:
\[
\left(\frac{x^6 + 1}{x^3}\right)^2
\]
Find the value of the given expression.

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Would you like to start with any particular problem, or should I solve them in order? Feel free to ask for more details on any specific step.

#### 5 Related Questions:
1. How do you solve quadratic equations involving radicals or fractions?
2. What is the general formula for solving \(x^3 - y^3\) given \(x + y\) and \(xy\)?
3. Can equations involving higher powers of \(x\) and \(y\) be simplified using symmetric identities?
4. How can you prove that certain algebraic identities hold true?
5. What methods are best for solving polynomial equations of higher degree?

**Tip**: When working with identities or fractions, try multiplying or factoring terms to simplify complex expressions.

The image contains multiple algebraic problems involving equations. The problems include solving for unknowns, proving algebraic identities, and working with powers and fractions.

This page provides solutions for algebraic equations involving powers, fractions, and identities, suitable for high school students (Grades 9-12). Learn how to solve for unknowns, prove identities, and manipulate algebraic expressions step by step.

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