Evaluate the Negative Exponent Expression: 9^-7

We are given the expression $9^{-7}$ and need to express it as a product of lower powers of 9 using negative exponents. This requires using the properties of exponents:

• If $a^m \times a^n = a^{m+n}$, then you can decompose a power into smaller factors.

To decompose $-7$, we seek integers whose sum equals $-7$. A possible set of integers is $-4$, $-2$, and $-1$. Therefore, we express $9^{-7}$ as:

$9^{-7} = 9^{-4} \times 9^{-2} \times 9^{-1}$

Looking at the given multiple-choice options:

• Option 1: $9^{-7} \times 9^{-1}$ would sum the exponents to become $-7 + (-1) = -8$, which is incorrect.
• Option 2: $9^{-4} \times 9^{-2} \times 9^{-1}$ correctly adds $-4 + (-2) + (-1) = -7$, matching our requirement.
• Option 3: $9^7 \times 9^{-1}$ simplifies to $7 - 1 = 6$, not $-7$, so it's incorrect.
• Option 4: "A+C are correct" is incorrect as neither A nor C solve the problem correctly.

Option 2 is correct. Therefore, the solution is:

The expression $9^{-7}$ can be written as $9^{-4} \times 9^{-2} \times 9^{-1}$.