The expression you are asking about is:

$\left( \frac{\sqrt{2}}{2} \right)^3$

Let's break it down step by step:

1. First, we simplify the expression inside the parentheses:

$\frac{\sqrt{2}}{2}$

2. Now, to cube the entire fraction, we apply the exponent to both the numerator and denominator:

$\left( \frac{\sqrt{2}}{2} \right)^3 = \frac{(\sqrt{2})^3}{2^3}$

3. Simplify each part:
   • $(\sqrt{2})^3 = 2^{3/2}$, which equals $2 \cdot \sqrt{2}$ (since $\sqrt{2} = 2^{1/2}$).
   • $2^3 = 8$.

So, the expression becomes:

$\frac{2\sqrt{2}}{8}$

4. Simplify the fraction:

$\frac{2\sqrt{2}}{8} = \frac{\sqrt{2}}{4}$

Final Answer:

$\left( \frac{\sqrt{2}}{2} \right)^3 = \frac{\sqrt{2}}{4}$

Would you like more details on any of the steps, or have any other questions?

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Related questions:

1. How would you simplify the cube of a different fraction?
2. What happens when you square a fraction with a square root in the numerator?
3. How do exponents work when applied to square roots?
4. What’s the difference between simplifying an expression with exponents versus multiplying them?
5. How can we express cube roots in terms of fractions?

Tip: When working with fractional exponents or roots, break down the calculation into smaller steps (like separating the numerator and denominator), and simplify each part.