https://file.aiquickdraw.com/mathbot/file/1726550349975bnvf2ps3.jpg

The given expression is \((x^2 y z^3)^5\). To simplify it, we can apply the power of a product rule, which states:

\[
(a \cdot b \cdot c)^n = a^n \cdot b^n \cdot c^n
\]

We'll distribute the exponent \(5\) to each part inside the parentheses:

1. For \(x^2\), we raise \(x^2\) to the power of 5:
   \[
   (x^2)^5 = x^{2 \cdot 5} = x^{10}
   \]
2. For \(y\), we raise \(y\) to the power of 5:
   \[
   y^1 \cdot 5 = y^5
   \]
3. For \(z^3\), we raise \(z^3\) to the power of 5:
   \[
   (z^3)^5 = z^{3 \cdot 5} = z^{15}
   \]

So the simplified expression is:
\[
x^{10} y^5 z^{15}
\]

It appears that in the image, the exponent for \(y\) was mistakenly written as 2. The correct final answer should be:

\[
x^{10} y^5 z^{15}
\]

Would you like further clarification on this? 

Here are 5 follow-up questions based on this explanation:
1. What is the power rule for exponents?
2. How do you simplify expressions involving exponents inside parentheses?
3. Can the exponent \(5\) apply to any other operations besides multiplication?
4. How would the result change if the exponent applied to a sum instead of a product?
5. What is the general rule for distributing exponents over multiple variables?

**Tip**: Always double-check the exponents for all variables when simplifying expressions to avoid small mistakes!

Learn how to simplify the exponential expression (x^2 y z^3)^5 using the power rule for exponents. This problem involves distributing exponents across multiple variables, covering key concepts such as exponentiation and power rules. Suitable for students in Grades 8-10.

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