Question

which expression is equivalent to \\(\frac{7^{-32}}{7^2}\\)? \\(7^{-30}\\) \\(\frac{1}{7^{-16}}\\) \\(7^{34}\\) \\(\frac{1}{7^{34}}\\)

Explanation:

Step1: Recall exponent rule for division

When dividing exponents with the same base, we subtract the exponents: $\frac{a^m}{a^n} = a^{m - n}$. Here, the base $a = 7$, $m = -32$, and $n = 2$.

Step2: Apply the exponent rule

Calculate $m - n$: $-32 - 2 = -34$. So $\frac{7^{-32}}{7^2} = 7^{-34}$.

Step3: Recall negative exponent rule

A negative exponent means the reciprocal: $a^{-n} = \frac{1}{a^n}$. So $7^{-34} = \frac{1}{7^{34}}$.

Answer:

$\frac{1}{7^{34}}$ (the bottom - right option)