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!