which expression is equivalent to $(2^{3})^{-5}$?\n$\frac{1}{2^{15}}$\n$\frac{1}{2^{8}}$\n$2^{8}$\n$2^{15}$

Explanation:

Step1: Apply power - of - a - power rule

According to the rule $(a^m)^n=a^{mn}$, for $(2^{3})^{-5}$, we have $a = 2$, $m = 3$, and $n=-5$. So $(2^{3})^{-5}=2^{3\times(-5)}$.

Step2: Calculate the exponent

$3\times(-5)=- 15$, so $2^{3\times(-5)}=2^{-15}$.

Step3: Use negative - exponent rule

The negative - exponent rule states that $a^{-n}=\frac{1}{a^{n}}$. For $2^{-15}$, we get $2^{-15}=\frac{1}{2^{15}}$.

Answer:

$\frac{1}{2^{15}}$