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.

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.