Multiply and Simplify: 9^-1 × 9^-2 × 9^-3 Using Exponent Rules

To solve the problem $9^{-1} \times 9^{-2} \times 9^{-3}$, we follow these steps:

• Step 1: Use the rule for multiplying exponential terms with the same base. The formula is $a^m \times a^n = a^{m+n}$.
• Step 2: Apply the formula to the given expression: $9^{-1} \times 9^{-2} \times 9^{-3} = 9^{-1-2-3}$.
• Step 3: Simplify the exponent: $-1 - 2 - 3 = -6$. Therefore, $9^{-1} \times 9^{-2} \times 9^{-3} = 9^{-6}$.

We can express $9^{-6}$ as a positive power by recalling that negative exponents indicate reciprocals:

$9^{-6} = \frac{1}{9^6}$.

Thus, both $9^{-6}$ and $\frac{1}{9^6}$ are valid expressions for the simplified form. Additionally, the expression $9^{-1-2-3}$ highlights the step where we combined the exponents, and it is equivalent to the final result. Therefore, all given answers correctly represent the simplified expression.

Therefore, the solution to the problem is All answers are correct.