Question
Rewrite the fol \[ \left(\frac{5}{3}\right)^{-2} \]
Ask by Mcguire Cook. in the United States
Jan 23,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\left(\frac{5}{3}\right)^{-2} = \frac{9}{25}
\]
Solution
Certainly! Let's rewrite the expression:
\[
\left(\frac{5}{3}\right)^{-2}
\]
### Step 1: Understand Negative Exponents
A negative exponent indicates that you take the reciprocal of the base and then apply the positive exponent. In general:
\[
\left(\frac{a}{b}\right)^{-n} = \left(\frac{b}{a}\right)^{n}
\]
### Step 2: Apply the Rule to the Given Expression
Applying this rule to your expression:
\[
\left(\frac{5}{3}\right)^{-2} = \left(\frac{3}{5}\right)^{2}
\]
### Step 3: Simplify the Exponent
Now, square both the numerator and the denominator:
\[
\left(\frac{3}{5}\right)^{2} = \frac{3^2}{5^2} = \frac{9}{25}
\]
### Final Answer
\[
\left(\frac{5}{3}\right)^{-2} = \frac{9}{25}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To rewrite \(\left(\frac{5}{3}\right)^{-2}\), we can apply the rule of negative exponents, which states that \(a^{-n} = \frac{1}{a^n}\). Therefore, we have: \[ \left(\frac{5}{3}\right)^{-2} = \frac{1}{\left(\frac{5}{3}\right)^{2}} \] Now, squaring the fraction gives us: \[ \left(\frac{5}{3}\right)^{2} = \frac{5^2}{3^2} = \frac{25}{9} \] Thus, we can rewrite our original expression: \[ \frac{1}{\left(\frac{5}{3}\right)^{2}} = \frac{1}{\frac{25}{9}} = \frac{9}{25} \] So, \(\left(\frac{5}{3}\right)^{-2}\) simplifies to \(\frac{9}{25}\).
