Question

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

Explanation:

Step1: Expand the numerator

Use power rule $(ab)^n=a^nb^n$ and $(a^m)^n=a^{mn}$
$(4r^{-1})^3 = 4^3 \cdot (r^{-1})^3 = 64r^{-3}$

Step2: Rewrite the expression

Substitute expanded numerator into the fraction
$\frac{64r^{-3}}{12r^{-2}}$

Step3: Simplify coefficients and exponents

Simplify $\frac{64}{12}=\frac{16}{3}$; use $\frac{a^m}{a^n}=a^{m-n}$ for exponents: $r^{-3-(-2)}=r^{-1}$
$\frac{16}{3}r^{-1}$

Step4: Convert to positive exponents

Use $a^{-n}=\frac{1}{a^n}$
$\frac{16}{3r}$

Answer:

$\frac{16}{3r}$