Question
Evaluate the exponential. \( \left(\frac{125}{8}\right)^{-\frac{2}{3}} \) Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. \( \left(\frac{125}{8}\right)^{-\frac{2}{3}}=\square \) (Type an integer or a simplified fraction.) B. The solution is not a real number.
Ask by Bartlett Goodwin. in the United States
Mar 12,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( \left(\frac{125}{8}\right)^{-\frac{2}{3}} = \frac{4}{25} \)
Solution
Calculate or simplify the expression \( \left(\frac{125}{8}\right)^{-\frac{2}{3}} \).
Calculate the value by following steps:
- step0: Calculate:
\(\left(\frac{125}{8}\right)^{-\frac{2}{3}}\)
- step1: Transform the expression:
\(\left(\frac{8}{125}\right)^{\frac{2}{3}}\)
- step2: Use the properties of exponents:
\(\frac{8^{\frac{2}{3}}}{125^{\frac{2}{3}}}\)
- step3: Evaluate the power:
\(\frac{4}{125^{\frac{2}{3}}}\)
- step4: Evaluate the power:
\(\frac{4}{25}\)
The value of \( \left(\frac{125}{8}\right)^{-\frac{2}{3}} \) is \( \frac{4}{25} \) or 0.16.
Therefore, the correct choice is:
A. \( \left(\frac{125}{8}\right)^{-\frac{2}{3}}=\frac{4}{25} \)
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Simplify this solution Bonus Knowledge
To evaluate \( \left(\frac{125}{8}\right)^{-\frac{2}{3}} \), we can first rewrite it as \( \frac{1}{\left(\frac{125}{8}\right)^{\frac{2}{3}}} \). Now, we calculate \( \left(\frac{125}{8}\right)^{\frac{2}{3}} \) which means taking the cube root of \( \frac{125}{8} \) and then squaring it. The cube root of \( \frac{125}{8} \) is \( \frac{5}{2} \), and squaring this gives us \( \left(\frac{5}{2}\right)^2 = \frac{25}{4} \). Finally, the original expression evaluates to \( \frac{1}{\frac{25}{4}} = \frac{4}{25} \). Thus, the answer is: A. \( \left(\frac{125}{8}\right)^{-\frac{2}{3}}=\frac{4}{25} \)
