Question

simpify: \\(\left( \dfrac{7x^{-2}y^{-5}}{14x^{3}y^{-9}} \
ight)^{-3}\\)

Explanation:

Step1: Simplify the coefficient and use exponent rules for division

First, simplify the coefficient \(\frac{7}{14}=\frac{1}{2}\). Then, for the variables, use the rule \(a^m\div a^n = a^{m - n}\). For \(x\): \(x^{-2}\div x^{3}=x^{-2 - 3}=x^{-5}\). For \(y\): \(y^{-5}\div y^{-9}=y^{-5-(-9)}=y^{4}\). So the expression inside the parentheses becomes \(\frac{1}{2}x^{-5}y^{4}\).

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

Now, raise each part to the power of \(-3\). For the coefficient: \((\frac{1}{2})^{-3}=2^{3}=8\). For \(x\): \((x^{-5})^{-3}=x^{15}\). For \(y\): \((y^{4})^{-3}=y^{-12}\).

Step3: Combine the results

Multiply the coefficient and the variables together: \(8x^{15}y^{-12}\). Then, rewrite \(y^{-12}\) as \(\frac{1}{y^{12}}\), so the simplified form is \(\frac{8x^{15}}{y^{12}}\).

Answer:

\(\boldsymbol{\frac{8x^{15}}{y^{12}}}\)