Question

which expression is equivalent to $\frac{4^{36}}{4^{-1}}$?
$\frac{1}{4^{-36}}$
$4^{-36}$
$4^{-37}$
$\frac{1}{4^{-37}}$

Explanation:

Step1: Apply exponent division rule

When dividing exponents with the same base, subtract the denominator exponent from the numerator exponent: $a^m \div a^n = a^{m-n}$.
$$\frac{4^{36}}{4^{-1}} = 4^{36 - (-1)}$$

Step2: Simplify the exponent

Calculate the value of the exponent by resolving the subtraction of a negative number.
$$4^{36 - (-1)} = 4^{36 + 1} = 4^{37}$$

Step3: Rewrite to match options

A positive exponent can be rewritten as the reciprocal of the same base with a negative exponent: $a^k = \frac{1}{a^{-k}}$.
$$4^{37} = \frac{1}{4^{-37}}$$

Answer:

$\frac{1}{4^{-37}}$