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Answer
\( (-27)^{-\frac{5}{3}} = -\frac{1}{243} \)
Solution
Calculate or simplify the expression \( (-27)^{-\frac{5}{3}} \).
Calculate the value by following steps:
- step0: Calculate:
\(\left(-27\right)^{-\frac{5}{3}}\)
- step1: Express with a positive exponent:
\(\frac{1}{\left(-27\right)^{\frac{5}{3}}}\)
- step2: Evaluate the power:
\(\frac{1}{-243}\)
- step3: Rewrite the fraction:
\(-\frac{1}{243}\)
The simplified form of \( (-27)^{-\frac{5}{3}} \) is \(-\frac{1}{243}\).
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Simplify this solution Beyond the Answer
To simplify \( (-27)^{-\frac{5}{3}} \), first consider the negative exponent. The expression can be rewritten as: \[ \frac{1}{(-27)^{\frac{5}{3}}} \] Next, we simplify \( (-27)^{\frac{5}{3}} \). The fractional exponent indicates that we take the cube root of -27 and then raise it to the 5th power. The cube root of -27 is -3, since: \[ (-3)^3 = -27 \] Now we raise -3 to the power of 5: \[ (-3)^5 = -243 \] Putting this back into our expression, we have: \[ (-27)^{-\frac{5}{3}} = \frac{1}{-243} \] Thus, the simplified result is: \[ -\frac{1}{243} \]
