evaluate each expression.\n$(25^{-\frac{3}{2}})^{\frac{1}{3}}=$

Explanation:

Step1: Use power - of - a - power rule

According to the rule $(a^m)^n=a^{mn}$, for $(25^{-\frac{3}{2}})^{\frac{1}{3}}$, we have $a = 25$, $m=-\frac{3}{2}$, and $n=\frac{1}{3}$. Then $(25^{-\frac{3}{2}})^{\frac{1}{3}}=25^{-\frac{3}{2}\times\frac{1}{3}}$. $25^{-\frac{3}{2}\times\frac{1}{3}}=25^{-\frac{1}{2}}$

Step2: Rewrite the negative exponent

Recall that $a^{-n}=\frac{1}{a^{n}}$. So $25^{-\frac{1}{2}}=\frac{1}{25^{\frac{1}{2}}}$.

Step3: Evaluate the square - root

Since $25^{\frac{1}{2}}=\sqrt{25}=5$, then $\frac{1}{25^{\frac{1}{2}}}=\frac{1}{5}$.

Answer:

$\frac{1}{5}$