Question

select all the expressions that are equivalent to $2^{-2} \cdot 7^{-2}$.
$14^{4}$
$\frac{1}{14^{2}}$
$14^{-4}$
$\frac{1}{14^{-2}}$
submit

Explanation:

Step1: Rewrite using exponent rule

Recall $a^{-n}=\frac{1}{a^n}$, so:
$2^{-2} \cdot 7^{-2} = \frac{1}{2^2} \cdot \frac{1}{7^2} = \frac{1}{2^2 \cdot 7^2}$

Step2: Apply product exponent rule

Use $(a \cdot b)^n = a^n \cdot b^n$, so $2^2 \cdot 7^2 = (2 \cdot 7)^2 = 14^2$. Substitute:
$\frac{1}{2^2 \cdot 7^2} = \frac{1}{14^2}$

Step3: Rewrite as negative exponent

Use $\frac{1}{a^n}=a^{-n}$, so:
$\frac{1}{14^2} = 14^{-2}$
Now check all options:

• $14^4$: Not equivalent.
• $\frac{1}{14^2}$: Equivalent (matches Step2 result).
• $14^{-4}$: Not equivalent.
• $\frac{1}{14^{-2}} = 14^2$: Not equivalent.

Answer:

$\frac{1}{14^2}$