Question

simplify.
\\(\left(\dfrac{5m^{-1}n^{6}}{3n^{-6}m^{-4}}\
ight)^{-2}\\)
write your answer using only positive exponents.

Explanation:

Step1: Simplify exponents inside the fraction

First, simplify the exponents of \(m\) and \(n\) in the fraction \(\frac{5m^{-1}n^{6}}{3n^{-6}m^{-4}}\). For \(m\): \(-1 - (-4)= -1 + 4 = 3\). For \(n\): \(6 - (-6)=6 + 6 = 12\). So the fraction becomes \(\frac{5m^{3}n^{12}}{3}\).

Step2: Apply the outer exponent \(-2\)

Now apply the exponent \(-2\) to \(\frac{5m^{3}n^{12}}{3}\). Using the power of a quotient rule \((\frac{a}{b})^n=\frac{a^n}{b^n}\) and power of a product rule \((ab)^n = a^n b^n\), we get \(\frac{(5)^{-2}(m^{3})^{-2}(n^{12})^{-2}}{(3)^{-2}}\).

Step3: Simplify each exponent

Simplify each exponent: \((5)^{-2}=\frac{1}{5^{2}}=\frac{1}{25}\), \((m^{3})^{-2}=m^{-6}=\frac{1}{m^{6}}\), \((n^{12})^{-2}=n^{-24}=\frac{1}{n^{24}}\), \((3)^{-2}=\frac{1}{3^{2}}=\frac{1}{9}\). So now we have \(\frac{\frac{1}{25}\cdot\frac{1}{m^{6}}\cdot\frac{1}{n^{24}}}{\frac{1}{9}}\).

Step4: Divide by a fraction (multiply by reciprocal)

Dividing by \(\frac{1}{9}\) is the same as multiplying by \(9\), so we get \(\frac{9}{25m^{6}n^{24}}\).

Answer:

\(\frac{9}{25m^{6}n^{24}}\)