Question

simplify. (-32)^{-\frac{8}{5}}

Explanation:

Step1: Rewrite -32 as a power

We know that $-32=-2^5$. So, $(-32)^{-\frac{8}{5}}=(-2^5)^{-\frac{8}{5}}$.

Step2: Apply power - of - a - power rule

According to the power - of - a - power rule $(a^m)^n=a^{mn}$. Here, $a = - 2$, $m = 5$, and $n=-\frac{8}{5}$. So, $(-2^5)^{-\frac{8}{5}}=(-2)^{5\times(-\frac{8}{5})}=(-2)^{-8}$.

Step3: Use negative exponent rule

The negative exponent rule states that $a^{-n}=\frac{1}{a^n}$. So, $(-2)^{-8}=\frac{1}{(-2)^8}$.

Step4: Calculate $(-2)^8$

$(-2)^8=256$ since an even power of a negative number is positive. So, $\frac{1}{(-2)^8}=\frac{1}{256}$.

Answer:

$\frac{1}{256}$