Question

select all the expressions that are equivalent to $8^{-8} \cdot 8^{-5}$.
$8^{-13}$
$\frac{1}{8^{-13}}$
$\frac{1}{8^{40}}$
$(8^{-4})^{-9}$

Explanation:

Step1: Apply exponent product rule

When multiplying exponents with the same base, add the exponents: $a^m \cdot a^n = a^{m+n}$.
$8^{-8} \cdot 8^{-5} = 8^{-8 + (-5)} = 8^{-13}$

Step2: Rewrite negative exponent

A negative exponent means the reciprocal: $a^{-n} = \frac{1}{a^n}$.
$8^{-13} = \frac{1}{8^{13}}$

Step3: Analyze other options

• $\frac{1}{8^{-13}} = 8^{13}$, which is not equal to $8^{-13}$.
• $\frac{1}{8^{40}} = 8^{-40}$, which is not equal to $8^{-13}$.
• $(8^{-4})^{-9} = 8^{(-4) \times (-9)} = 8^{36}$, which is not equal to $8^{-13}$.

Answer:

A. $8^{-13}$