Let's break down the expression \( x = (2 \cdot 2^{1/2})^3 \) step by step.

### Step 1: Simplify the inside of the parentheses
The term inside the parentheses is \( 2 \cdot 2^{1/2} \).

- \( 2^{1/2} \) is the square root of 2, which is approximately \( 1.414 \).
- So, \( 2 \cdot 2^{1/2} = 2 \cdot 1.414 = 2\sqrt{2} \).

### Step 2: Cube the result
Now, we need to cube the expression \( 2\sqrt{2} \).

- \( (2\sqrt{2})^3 = 2^3 \cdot (\sqrt{2})^3 = 8 \cdot 2^{3/2} \).

### Step 3: Simplify further
The term \( 2^{3/2} \) can be written as \( 2 \cdot \sqrt{2} \).

So, the entire expression becomes:
\[
x = 8 \cdot 2\sqrt{2} = 16\sqrt{2}.
\]

Thus, the simplified form of \( x \) is:
\[
x = 16\sqrt{2}.
\]

Let me know if you'd like more details or further clarifications!

Here are 5 related questions:
1. What is the cube root of \( 16\sqrt{2} \)?
2. How can you represent powers of 2 in different ways?
3. How do you simplify expressions involving square roots and exponents?
4. What is the value of \( 2^{1/2} \) in decimal form?
5. How would the expression change if the exponent were different?

**Tip**: When dealing with roots and exponents, always simplify the inner terms first before applying higher operations like squaring or cubing!

Learn how to simplify the expression (2 * 2^1/2)^3 step-by-step, involving exponents and square roots. This problem explores the laws of exponents and provides a detailed breakdown suitable for students in Grades 10-12.

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