Question

select all the expressions that are equivalent to $\frac{6^{-2}}{3^{-2}}$.2$2^{0}LXB0\frac{1}{2^{2}}$submit

Explanation:

Step1: Simplify negative exponents

Recall $a^{-n}=\frac{1}{a^n}$, so rewrite the expression:
$$\frac{6^{-2}}{3^{-2}} = \frac{\frac{1}{6^2}}{\frac{1}{3^2}}$$

Step2: Divide the fractions

Dividing by a fraction is multiplying by its reciprocal:
$$\frac{\frac{1}{6^2}}{\frac{1}{3^2}} = \frac{1}{6^2} \times 3^2 = \frac{3^2}{6^2}$$

Step3: Simplify the fraction

Note $6=2\times3$, so $6^2=(2\times3)^2=2^2\times3^2$:
$$\frac{3^2}{2^2\times3^2} = \frac{1}{2^2} = 4$$
Now evaluate each option:

• $2$: $2

eq4$, not equivalent

• $2^0$: $2^0=1

eq4$, not equivalent

• $\frac{1}{2^{-2}}$: $2^{-2}=\frac{1}{4}$, so $\frac{1}{\frac{1}{4}}=4$, equivalent
• $\frac{1}{2^2}$: $\frac{1}{4}

eq4$, not equivalent

Answer:

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