Which expressions are equivalent to $\frac{4^{-3}}{4^{-1}}$ ? $\newline$ Choose $\mathbf{2}$ answers: $\newline$ A $\frac{4^{1}}{4^{3}}$ $\newline$ B $\frac{1}{4^{2}}$ $\newline$ C $4^{3} \cdot 4^{1}$ $\newline$ D $\left(4^{-1}\right)^{-3}$

Full solution

Q. Which expressions are equivalent to

\frac{4^{-3}}{4^{-1}}

\newline

Choose

\mathbf{2}

answers:

\newline

\frac{4^{1}}{4^{3}}

\newline

\frac{1}{4^{2}}

\newline

4^{3} \cdot 4^{1}

\newline

\left(4^{-1}\right)^{-3}

Simplify Exponents: Simplify the given expression using the properties of exponents. $\newline$ When dividing powers with the same base, subtract the exponents. $\newline$ $(4^{-3})/(4^{-1}) = 4^{-3 - (-1)} = 4^{-3 + 1} = 4^{-2}$
Divide Powers: Compare the simplified expression with the given options. $\newline$ We have simplified the expression to $4^{-2}$ . Now we need to see which options are equivalent to this expression. $\newline$ Option A: $(4^{1})/(4^{3}) = 4^{1-3} = 4^{-2}$ $\newline$ Option B: $(1)/(4^{2}) = 4^{-2}$ $\newline$ Option C: $4^{3}*4^{1} = 4^{3+1} = 4^{4}$ , which is not equivalent to $4^{-2}$ . $\newline$ Option D: $(4^{-1})^{(-3)} = 4^{-1 * -3} = 4^{3}$ , which is not equivalent to $4^{-2}$ . $\newline$ Options A and B are equivalent to $4^{-2}$ .