https://file.aiquickdraw.com/mathbot/file/1727401973543z23jwdid.jpg

The image shows an expression in the form of a fraction raised to a power of 9. Here’s the breakdown of the expression:

\[
\left( \frac{e^4 f}{f^5 g h^7} \right)^9
\]

### Step 1: Simplify the expression inside the parentheses
First, simplify the expression inside the parentheses:

\[
\frac{e^4 f}{f^5 g h^7}
\]

We can simplify the powers of \( f \):
\[
\frac{e^4 f^1}{f^5 g h^7} = \frac{e^4}{f^{5-1} g h^7} = \frac{e^4}{f^4 g h^7}
\]

Now, the expression inside the parentheses simplifies to:
\[
\frac{e^4}{f^4 g h^7}
\]

### Step 2: Apply the exponent (9)
Now, raise the entire fraction to the 9th power:
\[
\left( \frac{e^4}{f^4 g h^7} \right)^9
\]

When raising a fraction to a power, we apply the exponent to both the numerator and the denominator:

\[
\frac{(e^4)^9}{(f^4 g h^7)^9} = \frac{e^{36}}{f^{36} g^9 h^{63}}
\]

### Final Answer:
\[
\frac{e^{36}}{f^{36} g^9 h^{63}}
\]

This is the fully simplified form of the given expression.

Would you like further clarification or details on any step? 

Here are five related questions to consider:
1. What are the rules for simplifying exponents when dividing powers with the same base?
2. How do you apply exponents to both the numerator and denominator of a fraction?
3. What are the key exponent rules used in algebraic simplifications?
4. How does raising a power to another power affect the exponent?
5. Can the simplified expression be factored further, or is it in its most reduced form?

**Tip:** Always simplify expressions inside parentheses before applying any outer exponents for easier calculations.

Discover how to simplify the algebraic expression (e^4 f / f^5 gh^7)^9 using exponent rules. This step-by-step solution demonstrates the application of power rules and the simplification of exponents. Suitable for Grades 9-11.

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