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.

This solution shows how to simplify the expression 1 / (3√(8^2)) / (1 / (3√(27^2))) step-by-step. It explains key algebraic concepts such as cube roots and fraction division. The problem is suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation