To show that \(\sqrt{5 + 2\sqrt{6}} = \sqrt{3} + \sqrt{2}\), let's start by squaring both sides and checking if they are equal.

### Step 1: Square both sides

We are given:
\[
\sqrt{5 + 2\sqrt{6}} = \sqrt{3} + \sqrt{2}
\]

Square both sides to remove the square roots:
\[
\left(\sqrt{5 + 2\sqrt{6}}\right)^2 = \left(\sqrt{3} + \sqrt{2}\right)^2
\]

On the left-hand side:
\[
\left(\sqrt{5 + 2\sqrt{6}}\right)^2 = 5 + 2\sqrt{6}
\]

On the right-hand side, use the binomial expansion formula \((a + b)^2 = a^2 + 2ab + b^2\):
\[
\left(\sqrt{3} + \sqrt{2}\right)^2 = \left(\sqrt{3}\right)^2 + 2\cdot\sqrt{3}\cdot\sqrt{2} + \left(\sqrt{2}\right)^2
\]

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

### Step 2: Compare both sides

The left-hand side is \(5 + 2\sqrt{6}\) and the right-hand side is also \(5 + 2\sqrt{6}\). Since both sides are equal, the original equation holds true.

Thus, we have shown that:
\[
\sqrt{5 + 2\sqrt{6}} = \sqrt{3} + \sqrt{2}
\]

Would you like more details or explanations on any step?

---

Here are some follow-up questions:

1. Can you square both sides of a more complex radical equation like this?
2. How do you simplify expressions with multiple square roots?
3. What is the importance of binomial expansion in solving this kind of problem?
4. How does this type of equation relate to solving quadratic equations?
5. Could you generalize this method for any expression involving nested square roots?

**Tip**: Always check your result by squaring both sides when solving radical equations—it helps verify your solution.

This detailed solution shows how to prove that √(5 + 2√6) = √3 + √2 by squaring both sides and applying the binomial expansion. Learn key concepts of radical expressions and algebra, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation