Question

Which expression is equivalent to (startfraction 4 m n over m superscript negative 2 baseline n superscript 6 baseline endfraction) superscript negative 2? assume m not-equals 0, n not-equals 0.

Answer 1

The equivalent expression to $\left(\frac{4mn}{m^{-2}n^6}\right)^{-2}$ is given by $=\mathbf{\frac{n^{10}}{16m^6}}$.

From the principle of indices we get,

Addition of exponent, $a^{m+n}=a^m.a^n$

Subtraction, $a^{m-n}=\frac{a^m}{a^n}$

Multiplication, $a^{mn}=(a^m)^n$

negative exponent, $a^{-m}=\frac{1}{a^m}$

Here in the problem given expression is $\left(\frac{4mn}{m^{-2}n^6}\right)^{-2}$

Calculating we get,

$\left(\frac{4mn}{m^{-2}n^6}\right)^{-2}$

$=\left(4m^{1-(-2)}n^{1-6}\right)^{-2}$, using addition and subtraction of indices formulae

$=\left(4m^3n^{-5}\right)^{-2}$

$=16^{-1}m^{-6}n^{10}$, using multiplication of indices

$=\frac{n^{10}}{16m^6}$, using negative exponent formula

Hence the equivalent required expression is $=\frac{n^{10}}{16m^6}$.

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