Question

simplify.
\frac{(4x^{-1}y^{-2})^{3}}{(x^{-4}y^{-6})^{-2}}
write your answer using only positive exponents.

Explanation:

Step1: Apply power of a product rule to numerator and denominator

For the numerator \((4x^{-1}y^{-2})^{3}\), we use \((ab)^n = a^n b^n\), so \(4^3(x^{-1})^3(y^{-2})^3\). For the denominator \((x^{-4}y^{-6})^{-2}\), we use the same rule: \((x^{-4})^{-2}(y^{-6})^{-2}\).
\[
\frac{4^3(x^{-1})^3(y^{-2})^3}{(x^{-4})^{-2}(y^{-6})^{-2}}
\]

Step2: Apply power of a power rule (\((a^m)^n=a^{mn}\))

Simplify the exponents: \((x^{-1})^3 = x^{-3}\), \((y^{-2})^3 = y^{-6}\), \((x^{-4})^{-2}=x^{8}\), \((y^{-6})^{-2}=y^{12}\), and \(4^3 = 64\).
\[
\frac{64x^{-3}y^{-6}}{x^{8}y^{12}}
\]

Step3: Apply quotient rule for exponents (\(\frac{a^m}{a^n}=a^{m - n}\))

For \(x\): \(x^{-3-8}=x^{-11}\), for \(y\): \(y^{-6 - 12}=y^{-18}\).
\[
64x^{-11}y^{-18}
\]

Step4: Convert negative exponents to positive ( \(a^{-n}=\frac{1}{a^n}\))

\[
\frac{64}{x^{11}y^{18}}
\]

Answer:

\(\dfrac{64}{x^{11}y^{18}}\)