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We are tasked with proving the following identity:

\[
\sqrt{5 + 2\sqrt{6}} = \sqrt{3} + \sqrt{2}
\]

### Proof Outline

We want to show that the left-hand side simplifies to the right-hand side. To do this, we can approach the equation by squaring both sides and checking if the expressions on both sides match.

#### Step 1: Square both sides of the equation

\[
\left( \sqrt{5 + 2\sqrt{6}} \right)^2 = \left( \sqrt{3} + \sqrt{2} \right)^2
\]

The left-hand side simplifies to:

\[
5 + 2\sqrt{6}
\]

The right-hand side expands as:

\[
\left( \sqrt{3} + \sqrt{2} \right)^2 = (\sqrt{3})^2 + 2\cdot\sqrt{3}\cdot\sqrt{2} + (\sqrt{2})^2
\]

Simplifying this:

\[
3 + 2\sqrt{6} + 2
\]

\[
= 5 + 2\sqrt{6}
\]

#### Step 2: Verify the equality

Both sides simplify to \(5 + 2\sqrt{6}\), so the original equation is valid.

\[
\sqrt{5 + 2\sqrt{6}} = \sqrt{3} + \sqrt{2}
\]

Thus, the equation is proven correct.

---

Would you like a more detailed breakdown of any of these steps?

Here are some related questions to explore:

1. How do you expand binomials involving square roots?
2. Why is squaring both sides of an equation a valid step?
3. What are the implications of simplifying radical expressions?
4. How do you check if a square root expression can be simplified further?
5. What are some other common methods for solving radical equations?

**Tip:** Always remember to check for extraneous solutions when squaring both sides of an equation, as this operation can introduce additional roots.

Learn how to prove that √(5 + 2√6) equals √3 + √2 using algebraic techniques and properties of square roots. A step-by-step solution involves squaring both sides and simplifying. This guide is suitable for high school students in Grades 10-12.

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