Select the equivalent expression.
\[
\sqrt[15]{\frac{1}{s^{6} \cdot s^{4}}}
\]
\( s^{-\frac{3}{2}} \)
\( s^{\frac{2}{3}} \)
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\( s^{\frac{3}{2}} \)
\( s^{-\frac{2}{3}} \)

Answer

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Answer

Apply the 15th root: \(\sqrt[15]{s^{-10}} = s^{\frac{-10}{15}} = s^{\frac{-2}{3}}\)

Steps

Step 1 ：Combine exponents in the denominator: \(\frac{1}{s^{6}\cdot s^{4}} = \frac{1}{s^{(6+4)}} = \frac{1}{s^{10}}\)

Step 2 ：Rewrite the fraction as a negative exponent: \(\frac{1}{s^{10}} = s^{-10}\)

Step 3 ：Apply the 15th root: \(\sqrt[15]{s^{-10}} = s^{\frac{-10}{15}} = s^{\frac{-2}{3}}\)