The problem presented in the image asks to calculate the missing value in the equation:

$\frac{1}{9 \times 9 \times 9 \times 9 \times 9} = 9^{\text{?}}$

We can solve this step by step.

Step 1: Simplify the left-hand side (LHS)

The LHS is a product of five 9's in the denominator: $9 \times 9 \times 9 \times 9 \times 9 = 9^5$ Thus, the equation becomes: $\frac{1}{9^5} = 9^{\text{?}}$

Step 2: Express the fraction as an exponent

We know that: $\frac{1}{9^5} = 9^{-5}$

Step 3: Set the exponents equal

Now the equation is: $9^{-5} = 9^{\text{?}}$

Since both sides have the same base (9), we can directly equate the exponents: $\text{?} = -5$

Conclusion

The missing value is $-5$.

Would you like more details or have any questions about this? Here are 5 related questions you might find interesting:

1. How would the equation change if the base were a different number, such as 8?
2. What is the general rule for converting fractions to negative exponents?
3. How would you solve this equation if there were more terms in the denominator?
4. How can you visualize this problem on a number line or through logarithms?
5. What would the exponent be if the expression on the right side were squared?

Tip: Negative exponents indicate the reciprocal of the base raised to the positive exponent.