To solve the expression 

\[
2^3 \times \left(\frac{3^3 \times 2^{-2}}{3^{-2} \times 2^4}\right)^3,
\]

we can follow these steps:

### Step 1: Simplify Inside the Parentheses

First, look at the expression inside the parentheses:

\[
\frac{3^3 \times 2^{-2}}{3^{-2} \times 2^4}.
\]

**Simplify the numerator:**

- \(3^3 = 27\)
- \(2^{-2} = \frac{1}{2^2} = \frac{1}{4}\)

Therefore, the numerator becomes \(27 \times \frac{1}{4} = \frac{27}{4}\).

**Simplify the denominator:**

- \(3^{-2} = \frac{1}{3^2} = \frac{1}{9}\)
- \(2^4 = 16\)

Therefore, the denominator becomes \(\frac{1}{9} \times 16 = \frac{16}{9}\).

Now, substitute back:

\[
\frac{3^3 \times 2^{-2}}{3^{-2} \times 2^4} = \frac{\frac{27}{4}}{\frac{16}{9}}.
\]

### Step 2: Divide the Fractions

\[
\frac{\frac{27}{4}}{\frac{16}{9}} = \frac{27}{4} \times \frac{9}{16} = \frac{27 \times 9}{4 \times 16} = \frac{243}{64}.
\]

### Step 3: Apply the Cube to the Fraction

Now, cube the result from inside the parentheses:

\[
\left(\frac{243}{64}\right)^3 = \frac{243^3}{64^3}.
\]

Calculating the powers:

- \(243^3 = (3^5)^3 = 3^{15}\)
- \(64^3 = (2^6)^3 = 2^{18}\)

So,

\[
\left(\frac{243}{64}\right)^3 = \frac{3^{15}}{2^{18}}.
\]

### Step 4: Multiply by \(2^3\)

Now, go back to the original expression:

\[
2^3 \times \frac{3^{15}}{2^{18}}.
\]

\[
= \frac{2^3 \times 3^{15}}{2^{18}}.
\]

### Step 5: Combine Exponents

\[
= \frac{3^{15}}{2^{18-3}} = \frac{3^{15}}{2^{15}}.
\]

### Final Answer

\[
\frac{3^{15}}{2^{15}} = \left(\frac{3}{2}\right)^{15}.
\]

Would you like more details or have any questions?

Here are five related questions to consider:

1. How would you simplify \(\left(\frac{4^2 \times 5^{-1}}{2^{-3} \times 5^3}\right)^2\)?
2. What is the result of \( \frac{6^3 \times 2^{-4}}{3^{-3} \times 2^2} \)?
3. How do you deal with negative exponents in complex fractions?
4. Can you express \( (a^3 \times b^{-2})^2 \times (a^{-1} \times b^4)^{-1} \) in a simplified form?
5. How would the approach change if the original problem had a base of zero for any of the exponents?

**Tip:** Remember, when working with exponents, handling the negative exponents first can simplify the process, making multiplication and division more straightforward.

Learn how to solve the exponential expression 2^3 × (3^3 × 2^-2)/(3^-2 × 2^4)^3 with a detailed step-by-step solution. This problem involves complex exponent rules, including negative exponents and fractional exponents. 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