Which expression is equivalent to left parenthesis, start fraction, 3, to the power minus 1 , divided by, 3, to the power minus 2 , end fraction, right parenthesis, to the power minus 1 , question mark(

Question

3
−2

3
−1

​
)
−1
?
Answer
Multiple Choice Answers
11
99
one ninth
9
1
​

one third
3
1
​

Answer 1

To simplify the expression \(\left(\frac{3^{-1}}{3^{-2}}\right)^{-1}\), we can use the properties of exponents.

1. Simplifying the Fraction: \[ \frac{3^{-1}}{3^{-2}} = 3^{-1 - (-2)} = 3^{-1 + 2} = 3^1 = 3 \]

2. Applying the Outer Exponent: Now we take \(3\) and raise it to the power of \(-1\): \[ 3^{-1} = \frac{1}{3} \]

Thus, the expression \(\left(\frac{3^{-1}}{3^{-2}}\right)^{-1}\) simplifies to \(\frac{1}{3}\).

Therefore, the correct answer is \( \frac{1}{3} \), which is equivalent to "one third".