https://file.aiquickdraw.com/mathbot/file/17266967866261muy1erf.jpg

The expression shown in the image is:

\[
\left( \frac{m^3 p^5}{n^7} \right)^6 \cdot \left( \frac{m^2 n^0 p^3}{m^4 n^2} \right)^3
\]

We need to simplify this.

### Step-by-Step Solution

#### First Expression:
\[
\left( \frac{m^3 p^5}{n^7} \right)^6
\]

Apply the power rule: \((a^b)^c = a^{b \cdot c}\).

\[
= \frac{(m^3)^6 (p^5)^6}{(n^7)^6} = \frac{m^{18} p^{30}}{n^{42}}
\]

#### Second Expression:
\[
\left( \frac{m^2 n^0 p^3}{m^4 n^2} \right)^3
\]

First simplify the inside expression:
\[
\frac{m^2 n^0 p^3}{m^4 n^2} = \frac{p^3}{m^2 n^2}
\]

Now apply the power rule:

\[
\left( \frac{p^3}{m^2 n^2} \right)^3 = \frac{(p^3)^3}{(m^2)^3 (n^2)^3} = \frac{p^9}{m^6 n^6}
\]

#### Multiply Both Simplified Expressions:

Now multiply the two simplified expressions:

\[
\frac{m^{18} p^{30}}{n^{42}} \cdot \frac{p^9}{m^6 n^6} = \frac{m^{18} p^{30} p^9}{m^6 n^{42} n^6}
\]

Combine like terms:

\[
= \frac{m^{18 - 6} p^{30 + 9}}{n^{42 + 6}} = \frac{m^{12} p^{39}}{n^{48}}
\]

### Final Answer:
\[
\frac{m^{12} p^{39}}{n^{48}}
\]

Would you like further explanation or have any questions?

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Here are five related questions:
1. How would the expression change if the exponents were negative?
2. Can you simplify \(\left( \frac{x^2 y^3}{z^4} \right)^5\)?
3. What is the rule for multiplying powers with the same base?
4. How do you simplify an expression raised to a power inside a fraction?
5. How do you handle variables with zero exponents in expressions?

**Tip:** Always apply power rules carefully and combine terms with the same base to avoid mistakes.

Simplify the expression: (m^3 * p^5 / n^7)^6 * (m^2 * n^0 * p^3 / m^4 * n^2)^3

Learn how to simplify a complex expression with exponents involving variables m, n, and p. Apply the rules for powers, multiplication, and division of exponents to simplify the expression step-by-step. This example demonstrates key algebraic techniques suitable for Grades 8-10.

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