 2sqrt(2-sqrt3)(2+sqrt3) = (sqrt2) + (sqrt6)

Let's analyze and solve the equation step by step.

We have the expression:
\[
2\sqrt{2 - \sqrt{3}}(2 + \sqrt{3}) = \sqrt{2} + \sqrt{6}
\]

### Step 1: Expand the left-hand side (LHS)
We will multiply the term \( 2\sqrt{2 - \sqrt{3}} \) with \( (2 + \sqrt{3}) \). To do this, let's simplify \( \sqrt{2 - \sqrt{3}} \) first.

But before expanding it, let's assume this expression can be transformed into the right-hand side. First, we check if multiplying these terms will eventually lead to the simplified form of \( \sqrt{2} + \sqrt{6} \).

\[
2\sqrt{2 - \sqrt{3}}(2 + \sqrt{3})
\]

### Step 2: Assume the simplification of the product
We explore whether simplifying this product leads to \( \sqrt{2} + \sqrt{6} \).

I will calculate and simplify this step to confirm if this equality holds.

Let me perform the calculation.Upon simplifying both sides:

- The left-hand side (LHS) remains as \( 2\sqrt{2 - \sqrt{3}}(2 + \sqrt{3}) \).
- The right-hand side (RHS) is \( \sqrt{2} + \sqrt{6} \).

These two expressions are not equal. Therefore, the equation:
\[
2\sqrt{2 - \sqrt{3}}(2 + \sqrt{3}) = \sqrt{2} + \sqrt{6}
\]
is **false**.

Would you like more details on the simplification or have any further questions?

Here are 5 related questions to expand your understanding:
1. How do you simplify expressions involving nested square roots?
2. How do you factor expressions like \( \sqrt{a - b} \)?
3. What methods are used to check if two expressions are equivalent?
4. Can every square root expression be simplified neatly?
5. What strategies are used to solve radical equations?

**Tip:** Always try breaking down nested square roots first when simplifying complex radical expressions!

2sqrt(2-sqrt3)(2+sqrt3) = (sqrt2) + (sqrt6)

Discover how to simplify complex radical expressions like 2sqrt(2-sqrt3)(2+sqrt3) and compare it to sqrt(2) + sqrt(6). This step-by-step solution explores algebraic simplification techniques and radical expression equivalence, ideal for Grades 9-12 students.

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