We are tasked with simplifying the expression:

[ \frac{1}{\sqrt[3]{8^2}} \div \frac{1}{\sqrt[3]{27^2}} ]

Step 1: Simplify the cube roots

We start by evaluating the cube roots of the numbers inside the expression:

[ \sqrt[3]{8^2} = \sqrt[3]{64} = 4 \quad \text{(since } 4^3 = 64\text{)} ] [ \sqrt[3]{27^2} = \sqrt[3]{729} = 9 \quad \text{(since } 9^3 = 729\text{)} ]

Step 2: Substitute the values into the original expression

Now the expression becomes:

$\frac{1}{4} \div \frac{1}{9}$

Step 3: Simplify the division of fractions

Dividing by a fraction is the same as multiplying by its reciprocal. So we have:

$\frac{1}{4} \times \frac{9}{1} = \frac{9}{4}$

Thus, the simplified expression is:

$\frac{9}{4}$

Would you like further details or any clarification?

Here are some related questions you might find helpful:

1. How do cube roots work in algebraic expressions?
2. Can you explain how to simplify fraction division further?
3. What is the difference between square roots and cube roots?
4. How can you express the result $\frac{9}{4}$ as a decimal?
5. How do fractional exponents relate to cube roots?

Tip: When dividing fractions, remember to multiply by the reciprocal of the divisor to simplify efficiently.