What is expand x^{3/2}+x^{-1/3} ?

The solution to expand x^{3/2}+x^{-1/3} is 1/(x^{1/3)}+x^{3/2}

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expand x^{3/2}+x^{-1/3}

expand

x^{\frac{3}{2}} + x^{- \frac{1}{3}}

\frac{1}{x ^{\frac{1}{3}}} + x^{\frac{3}{2}}

Solution steps

x^{\frac{3}{2}} + x^{- \frac{1}{3}}

Apply exponent rule:

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

= x^{\frac{3}{2}} + \frac{1}{x ^{\frac{1}{3}}}

Convert element to fraction:

x^{\frac{3}{2}} = \frac{x ^{\frac{3}{2}} x ^{\frac{1}{3}}}{x ^{\frac{1}{3}}}

= \frac{1}{x ^{\frac{1}{3}}} + \frac{x ^{\frac{3}{2}} x ^{\frac{1}{3}}}{x ^{\frac{1}{3}}}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 + x ^{\frac{3}{2}} x ^{\frac{1}{3}}}{x ^{\frac{1}{3}}}

1 + x^{\frac{3}{2}} x^{\frac{1}{3}} = 1 + x^{\frac{3}{2} + \frac{1}{3}}

= \frac{1 + x ^{\frac{3}{2} + \frac{1}{3}}}{x ^{\frac{1}{3}}}

Join

\frac{3}{2} + \frac{1}{3} : \frac{11}{6}

= \frac{1 + x ^{\frac{11}{6}}}{x ^{\frac{1}{3}}}

Apply the fraction rule:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{1 + x ^{\frac{11}{6}}}{x ^{\frac{1}{3}}} = \frac{1}{x ^{\frac{1}{3}}} + \frac{x ^{\frac{11}{6}}}{x ^{\frac{1}{3}}}

= \frac{1}{x ^{\frac{1}{3}}} + \frac{x ^{\frac{11}{6}}}{x ^{\frac{1}{3}}}

Cancel

\frac{x ^{\frac{11}{6}}}{x ^{\frac{1}{3}}} : x^{\frac{3}{2}}

= \frac{1}{x ^{\frac{1}{3}}} + x^{\frac{3}{2}}

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\frac{2 z ( z ^{2} + 2 z - 8 )}{( z - 1 ) ( z + 2 )}

\frac{e ^{x}}{( 1 - e ^{x} ) ^{2} d x}

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\frac{h ( α - ρ )}{α}

What is expand x^{3/2}+x^{-1/3} ?
The solution to expand x^{3/2}+x^{-1/3} is 1/(x^{1/3)}+x^{3/2}