https://file.aiquickdraw.com/mathbot/file/1729471674884qhymodeo.jpg

The expression shown in the image is:

\[
\left( \frac{x^{-6}}{2w^5} \right)^{-3}
\]

### Step-by-step simplification:

1. **Apply the exponent to both the numerator and the denominator:**

   \[
   \left( \frac{x^{-6}}{2w^5} \right)^{-3} = \frac{\left( x^{-6} \right)^{-3}}{\left( 2w^5 \right)^{-3}}
   \]

2. **Simplify each part:**

   - For the numerator: \(\left( x^{-6} \right)^{-3} = x^{18}\), since \((a^m)^n = a^{m \cdot n}\).
   - For the denominator: \(\left( 2w^5 \right)^{-3} = 2^{-3}w^{-15} = \frac{1}{2^3 w^{15}} = \frac{1}{8w^{15}}\).

3. **Simplify the entire fraction:**

   \[
   \frac{x^{18}}{\frac{1}{8w^{15}}} = x^{18} \cdot \frac{8w^{15}}{1} = 8x^{18}w^{15}
   \]

### Final simplified expression:

\[
8x^{18}w^{15}
\]

Would you like any further details or have any questions?

### Related Questions:
1. How would the result change if the exponent were \(-2\) instead of \(-3\)?
2. Can you explain the general rule for handling negative exponents in fractions?
3. How would you approach the problem if the denominator contained a variable with a negative exponent?
4. How would this expression change if there were additional variables in the denominator?
5. What happens if the expression inside the parentheses was a polynomial instead of a monomial?

### Tip:
Always remember that a negative exponent "flips" the base when simplifying expressions, whether it's part of a fraction or a single term.

Learn how to simplify the expression (x^(-6) / (2w^5))^(-3) using exponentiation rules. This step-by-step guide covers the laws of exponents, including the power of a power and negative exponents. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation