Question

select all the expressions that are equivalent to $8^{-5} \cdot 7^{-5}$.$\frac{1}{56^{-5}}$$56^{-1} \cdot 56^{-4}$$56^{25}$$56^{-10}$save answer

Explanation:

Step1: Apply exponent product rule

Recall $a^n \cdot b^n=(a\cdot b)^n$. Here $a=8, b=7, n=-5$:
$8^{-5} \cdot 7^{-5}=(8 \times 7)^{-5}=56^{-5}$

Step2: Simplify option 1

Use $a^{-n}=\frac{1}{a^n}$, so $\frac{1}{56^{-5}}=56^{5}$. This does not match $56^{-5}$.

Step3: Simplify option 2

Use $a^m \cdot a^n=a^{m+n}$:
$56^{-1} \cdot 56^{-4}=56^{-1+(-4)}=56^{-5}$. This matches.

Step4: Analyze option 3

$56^{25}$ has a positive exponent 25, which does not match $56^{-5}$.

Step5: Analyze option 4

$56^{-10}$ has exponent -10, which does not match $56^{-5}$.

Answer:

$56^{-1} \cdot 56^{-4}$