Question

express the following fraction in simplest form, only using positive exponents.
$\frac{(4r^{-1})^{3}}{12r^{-2}}$

Explanation:

Step1: Expand numerator via exponent rule

$(4r^{-1})^3 = 4^3 \cdot (r^{-1})^3 = 64r^{-3}$

Step2: Rewrite the entire fraction

$\frac{64r^{-3}}{12r^{-2}}$

Step3: Simplify coefficient and exponents

Simplify $\frac{64}{12} = \frac{16}{3}$; use $r^a/r^b = r^{a-b}$: $r^{-3 - (-2)} = r^{-1} = \frac{1}{r}$

Step4: Combine simplified terms

$\frac{16}{3} \cdot \frac{1}{r} = \frac{16}{3r}$

Answer:

$\frac{16}{3r}$