Let's solve the expression \((2 - \sqrt{2})^2 \times (2 + \sqrt{2})^2\).

### Step 1: Expand each square term

First, expand \((2 - \sqrt{2})^2\) and \((2 + \sqrt{2})^2\).

\[
(2 - \sqrt{2})^2 = 2^2 - 2 \cdot 2 \cdot \sqrt{2} + (\sqrt{2})^2 = 4 - 4\sqrt{2} + 2 = 6 - 4\sqrt{2}
\]

\[
(2 + \sqrt{2})^2 = 2^2 + 2 \cdot 2 \cdot \sqrt{2} + (\sqrt{2})^2 = 4 + 4\sqrt{2} + 2 = 6 + 4\sqrt{2}
\]

### Step 2: Multiply the results

Now, multiply \((6 - 4\sqrt{2})\) by \((6 + 4\sqrt{2})\).

This expression takes the form \((a - b)(a + b)\), which is a difference of squares:

\[
(6 - 4\sqrt{2})(6 + 4\sqrt{2}) = 6^2 - (4\sqrt{2})^2
\]

Calculate each term:

\[
6^2 = 36
\]
\[
(4\sqrt{2})^2 = 16 \times 2 = 32
\]

So,

\[
36 - 32 = 4
\]

### Final Answer

\[
(2 - \sqrt{2})^2(2 + \sqrt{2})^2 = 4
\]

Would you like more details, or do you have any questions?

---

Here are 8 related questions:
1. How do you expand the square of a binomial?
2. What is the difference of squares formula?
3. How would the problem change if \(\sqrt{2}\) were replaced with another square root?
4. What is the geometric interpretation of the result?
5. How can you factor expressions involving square roots?
6. What happens if you cube the binomial instead of squaring it?
7. How does the difference of squares help simplify algebraic expressions?
8. Can you generalize this process to any binomial with irrational numbers?

**Tip:** Recognizing patterns like the difference of squares can greatly simplify algebraic expressions and speed up calculations!

Explore the solution to the problem (2-√2)^2(2+√2)^2 with a detailed step-by-step explanation. Understand the algebraic process involving binomial expansion and the difference of squares. Suitable for 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