https://file.aiquickdraw.com/mathbot/file/1726836491851kamfp6cj.jpg

solve these problems. (assume that any variables in the expression are nonzero)

I can see the image, and the two problems are as follows:

**Problem 11:**
\[
\left(\frac{3x}{y^5}\right)^{-5}
\]
You need to rewrite this using only positive exponents and simplify.

**Problem 12:**
\[
\frac{8y^{-1}z^{-5}}{16yz^{-5}}
\]
Again, this needs to be rewritten using positive exponents and simplified.

I'll solve both problems step by step.

### Problem 11 Solution:

We start by applying the negative exponent to both the numerator and the denominator of the fraction:
\[
\left(\frac{3x}{y^5}\right)^{-5} = \frac{(y^5)^5}{(3x)^5} = \frac{y^{25}}{3^5 \cdot x^5} = \frac{y^{25}}{243x^5}
\]

### Problem 12 Solution:

We simplify by combining like terms and eliminating negative exponents:
\[
\frac{8y^{-1}z^{-5}}{16yz^{-5}} = \frac{8}{16} \cdot \frac{y^{-1}}{y} \cdot \frac{z^{-5}}{z^{-5}} = \frac{1}{2} \cdot \frac{1}{y^2} \cdot 1 = \frac{1}{2y^2}
\]

Let me know if you need more clarification on these steps or further details!

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Here are 5 related questions to deepen your understanding:
1. How do you simplify expressions with negative exponents?
2. What happens when you multiply two powers with the same base?
3. How would you simplify \(\left(\frac{a}{b^2}\right)^{-3}\)?
4. What is the general rule for raising a fraction to a negative exponent?
5. How does changing the sign of an exponent affect the reciprocal of a fraction?

**Tip:** Always remember to convert negative exponents into positive ones by using reciprocals.

Rewrite the expression using only positive exponents, and simplify: (3x / y^5)^-5 and (8y^-1z^-5) / (16yz^-5).

Learn how to simplify expressions with negative exponents, such as (3x / y^5)^-5 and (8y^-1z^-5) / (16yz^-5), step-by-step. This problem covers algebraic concepts like exponent rules and simplifying expressions, suitable for students in Grades 9-12.

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