https://file.aiquickdraw.com/mathbot/file/1729631482062ujplsrcm.jpg

The equation presented in the image is:

\[
\frac{9}{9 \times 9 \times 9 \times 9 \times 9} = 9^f
\]

Let's break this down step by step.

### Step 1: Simplify the left-hand side.
The denominator consists of five 9's multiplied together, which is \( 9^5 \). So the equation becomes:

\[
\frac{9}{9^5} = 9^f
\]

### Step 2: Simplify the fraction.
We can rewrite the fraction \(\frac{9}{9^5}\) as:

\[
9^{1} \div 9^5 = 9^{1-5} = 9^{-4}
\]

So now the equation becomes:

\[
9^{-4} = 9^f
\]

### Step 3: Compare the exponents.
Since the bases are the same (9), we can equate the exponents:

\[
-4 = f
\]

Thus, the value of \( f \) is:

\[
\boxed{-4}
\]

Would you like more details on any step or have further questions? Here are some related questions:

1. How do exponent rules like \( a^m / a^n = a^{m-n} \) work in general?
2. What happens when the base in exponential equations is different?
3. How can we solve more complex equations involving fractional exponents?
4. What are the key properties of exponents that are helpful in algebra?
5. Can logarithms be used to solve exponential equations like this?

**Tip:** Always simplify the terms on both sides of an equation to make the solution more straightforward!

What is the value of f in the equation 9 / (9 × 9 × 9 × 9 × 9) = 9^f?

This problem demonstrates how to find the value of the exponent f in the equation 9 / (9 × 9 × 9 × 9 × 9) = 9^f using the laws of exponents. Ideal for students in Grades 8-10, this example highlights simplification techniques and exponentiation rules.

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