What is ((a/b)^{-2})^{3/4} ?

The solution to ((a/b)^{-2})^{3/4} is ((b^3)/(a^3))^{1/2}

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simplify ((a/b)^{-2})^{3/4}

simplify

((\frac{a}{b})^{- 2})^{\frac{3}{4}}

(\frac{b ^{3}}{a ^{3}})^{\frac{1}{2}}

Solution steps

((\frac{a}{b})^{- 2})^{\frac{3}{4}}

Apply exponent rule:

(a^{b})^{c} = a^{b c},

assuming

a \geq 0

= (\frac{a}{b})^{- 2 \cdot \frac{3}{4}}

- 2 \cdot \frac{3}{4} = - \frac{3}{2}

= (\frac{a}{b})^{- \frac{3}{2}}

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

(\frac{a}{b})^{- \frac{3}{2}} = \frac{1}{( \frac{a}{b} ) ^{\frac{3}{2}}}

= \frac{1}{( \frac{a}{b} ) ^{\frac{3}{2}}}

Simplify

(\frac{a}{b})^{\frac{3}{2}} : (\frac{a ^{3}}{b ^{3}})^{\frac{1}{2}}

= \frac{1}{( \frac{a ^{3}}{b ^{3}} ) ^{\frac{1}{2}}}

(\frac{a ^{3}}{b ^{3}})^{\frac{1}{2}} = \frac{a ^{\frac{3}{2}}}{b ^{\frac{3}{2}}}

= \frac{1}{\frac{a ^{\frac{3}{2}}}{b ^{\frac{3}{2}}}}

Apply the fraction rule:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{b ^{\frac{3}{2}}}{a ^{\frac{3}{2}}}

Combine same powers :

\frac{x ^{\frac{a}{m}}}{y ^{\frac{b}{m}}} = (\frac{x ^{a}}{y ^{b}})^{\frac{1}{m}}

= (\frac{b ^{3}}{a ^{3}})^{\frac{1}{2}}

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What is ((a/b)^{-2})^{3/4} ?
The solution to ((a/b)^{-2})^{3/4} is ((b^3)/(a^3))^{1/2}