https://file.aiquickdraw.com/mathbot/file/1725584531198ohccquwo.jpg

The expression in the image is:

\[
(2r^4)^{-5}
\]

Let's simplify this expression.

### Step 1: Apply the negative exponent rule
The negative exponent rule states that \(a^{-n} = \frac{1}{a^n}\). So, applying this rule:

\[
(2r^4)^{-5} = \frac{1}{(2r^4)^5}
\]

### Step 2: Apply the power of a product rule
The power of a product rule states that \((ab)^n = a^n \cdot b^n\). Applying this rule to the expression:

\[
\frac{1}{(2r^4)^5} = \frac{1}{2^5 \cdot (r^4)^5}
\]

### Step 3: Simplify each term
Now, calculate \(2^5\) and \((r^4)^5\):

- \(2^5 = 32\)
- \((r^4)^5 = r^{4 \times 5} = r^{20}\)

So, the expression simplifies to:

\[
\frac{1}{32r^{20}}
\]

Thus, the final simplified expression is:

\[
\boxed{\frac{1}{32r^{20}}}
\]

Would you like further details on any part of this solution or have any questions?

### Related Questions:
1. How do you simplify an expression with a negative exponent?
2. What is the power of a product rule in exponents?
3. How do you calculate the power of a number, such as \(2^5\)?
4. What are the steps to multiply exponents when raising a power to another power?
5. How does the negative exponent affect the position of the base in a fraction?

**Tip:** Always remember that a negative exponent indicates the reciprocal of the base raised to the positive of that exponent.

Learn how to simplify the exponent expression (2r^4)^{-5} using the negative exponent rule and power of a product rule. This step-by-step solution walks through applying key exponent rules to simplify the expression to \frac{1}{32r^{20}}. Suitable for students in Grades 9-10.

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