https://file.aiquickdraw.com/mathbot/file/1725497243989gd9khsiv.jpg

The expression in the image is:

\[
\frac{(16x^{-4})^2}{(4x^{-2})^3}
\]

Let's simplify this step-by-step.

### Step 1: Apply the power rule \((a^m)^n = a^{m \cdot n}\)
For the numerator:
\[
(16x^{-4})^2 = 16^2 \cdot x^{-4 \cdot 2} = 256x^{-8}
\]

For the denominator:
\[
(4x^{-2})^3 = 4^3 \cdot x^{-2 \cdot 3} = 64x^{-6}
\]

Now the expression becomes:
\[
\frac{256x^{-8}}{64x^{-6}}
\]

### Step 2: Simplify the constants
\[
\frac{256}{64} = 4
\]

So the expression is now:
\[
4 \cdot \frac{x^{-8}}{x^{-6}}
\]

### Step 3: Apply the quotient rule for exponents \(\frac{a^m}{a^n} = a^{m-n}\)
\[
\frac{x^{-8}}{x^{-6}} = x^{-8 - (-6)} = x^{-8 + 6} = x^{-2}
\]

Thus, the expression simplifies to:
\[
4x^{-2}
\]

### Final Answer:
\[
4x^{-2}
\]

If you'd like, I can explain any step in further detail.

Would you like more details, or do you have any questions?

Here are 5 questions to consider:
1. How do power rules apply to negative exponents?
2. Can you explain the difference between multiplying and dividing powers?
3. How does raising a term to a power affect both the base and exponent?
4. Why does \(x^{-2}\) represent a reciprocal, and what is its equivalent?
5. What happens when both the numerator and denominator have negative exponents?

**Tip:** When simplifying expressions with exponents, always apply exponent rules methodically to avoid mistakes.

Learn how to simplify the expression (16x^(-4))^2 / (4x^(-2))^3 step-by-step by applying the power and quotient rules for exponents. This problem is suitable for students in Grades 9-11 and covers key algebraic concepts like exponentiation and expression simplification.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation