Question
17. \( \left(9^{-3 / 5} \cdot 9^{1 / 5}\right)^{-1} \)
Ask by Rodriquez Powell. in the United States
Jan 21,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The simplified form is \( \sqrt[5]{81} \).
Solution
Calculate or simplify the expression \( (9^{-3/5} * 9^{1/5})^{-1} \).
Calculate the value by following steps:
- step0: Calculate:
\(\left(9^{\frac{-3}{5}}\times 9^{\frac{1}{5}}\right)^{-1}\)
- step1: Rewrite the fraction:
\(\left(9^{-\frac{3}{5}}\times 9^{\frac{1}{5}}\right)^{-1}\)
- step2: Multiply the numbers:
\(\left(9^{-\frac{2}{5}}\right)^{-1}\)
- step3: Transform the expression:
\(9^{-\frac{2}{5}\left(-1\right)}\)
- step4: Use the rules for multiplication and division:
\(9^{\frac{2}{5}}\)
- step5: Transform the expression:
\(\sqrt[5]{9^{2}}\)
- step6: Simplify:
\(\sqrt[5]{81}\)
The simplified form of \( \left(9^{-3 / 5} \cdot 9^{1 / 5}\right)^{-1} \) is \( \sqrt[5]{81} \).
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To simplify \( \left(9^{-3 / 5} \cdot 9^{1 / 5}\right)^{-1} \), first combine the exponents of 9 using the property that \( a^m \cdot a^n = a^{m+n} \): \[ 9^{-3/5} \cdot 9^{1/5} = 9^{(-3/5 + 1/5)} = 9^{-2/5} \] Now, apply the exponent on the outside: \[ \left(9^{-2/5}\right)^{-1} = 9^{2/5} \] Since \( 9 \) can be written as \( 3^2 \), we have: \[ 9^{2/5} = (3^2)^{2/5} = 3^{(2 \cdot 2/5)} = 3^{4/5} \] Thus, the simplified expression is: \[ \boxed{3^{4/5}} \]
