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Popular Algebra >

expand 4/3 pi((3a^2b^4-2a^4b)/2)^3

Solution

expand

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

Solution

- \frac{4 π a ^{12} b ^{3}}{3} + 6 π a^{10} b^{6} - 9 π a^{8} b^{9} + \frac{9 π a ^{6} b ^{12}}{2}

Solution steps

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

(\frac{3 a ^{2} b ^{4} - 2 a ^{4} b}{2})^{3} = \frac{27 a ^{6} b ^{12} - 54 a ^{8} b ^{9} + 36 a ^{10} b ^{6} - 8 a ^{12} b ^{3}}{2 ^{3}}

= π \frac{4}{3} \cdot \frac{- 8 a ^{12} b ^{3} + 36 a ^{10} b ^{6} - 54 a ^{8} b ^{9} + 27 a ^{6} b ^{12}}{2 ^{3}}

Multiply fractions:

a \cdot \frac{b}{c} \cdot \frac{d}{e} = \frac{a \cdot b \cdot d}{c \cdot e}

= \frac{4 ( 27 a ^{6} b ^{12} - 54 a ^{8} b ^{9} + 36 a ^{10} b ^{6} - 8 a ^{12} b ^{3} ) π}{3 \cdot 2 ^{3}}

Factor

27 a^{6} b^{12} - 54 a^{8} b^{9} + 36 a^{10} b^{6} - 8 a^{12} b^{3} : b^{3} a^{6} (27 b^{9} - 54 a^{2} b^{6} + 36 a^{4} b^{3} - 8 a^{6})

= \frac{4 b ^{3} a ^{6} ( 27 b ^{9} - 54 a ^{2} b ^{6} + 36 a ^{4} b ^{3} - 8 a ^{6} ) π}{3 \cdot 2 ^{3}}

Factor

4 : 2^{2}

= \frac{2 ^{2} π a ^{6} b ^{3} ( - 8 a ^{6} + 36 a ^{4} b ^{3} - 54 a ^{2} b ^{6} + 27 b ^{9} )}{2 ^{3} \cdot 3}

Cancel

\frac{2 ^{2} b ^{3} a ^{6} ( 27 b ^{9} - 54 a ^{2} b ^{6} + 36 a ^{4} b ^{3} - 8 a ^{6} ) π}{3 \cdot 2 ^{3}} : \frac{π a ^{6} b ^{3} ( - 8 a ^{6} + 36 a ^{4} b ^{3} - 54 a ^{2} b ^{6} + 27 b ^{9} )}{3 \cdot 2}

= \frac{π a ^{6} b ^{3} ( - 8 a ^{6} + 36 a ^{4} b ^{3} - 54 a ^{2} b ^{6} + 27 b ^{9} )}{3 \cdot 2}

Multiply the numbers:

3 \cdot 2 = 6

= \frac{π a ^{6} b ^{3} ( - 8 a ^{6} + 36 a ^{4} b ^{3} - 54 a ^{2} b ^{6} + 27 b ^{9} )}{6}

Expand

π a^{6} b^{3} (- 8 a^{6} + 36 a^{4} b^{3} - 54 a^{2} b^{6} + 27 b^{9}) : - 8 π a^{12} b^{3} + 36 π a^{10} b^{6} - 54 π a^{8} b^{9} + 27 π a^{6} b^{12}

= \frac{- 8 π a ^{12} b ^{3} + 36 π a ^{10} b ^{6} - 54 π a ^{8} b ^{9} + 27 π a ^{6} b ^{12}}{6}

Apply the fraction rule:

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

\frac{- 8 π a ^{12} b ^{3} + 36 π a ^{10} b ^{6} - 54 π a ^{8} b ^{9} + 27 π a ^{6} b ^{12}}{6} = - \frac{8 π a ^{12} b ^{3}}{6} + \frac{36 π a ^{10} b ^{6}}{6} - \frac{54 π a ^{8} b ^{9}}{6} + \frac{27 π a ^{6} b ^{12}}{6}

= - \frac{8 π a ^{12} b ^{3}}{6} + \frac{36 π a ^{10} b ^{6}}{6} - \frac{54 π a ^{8} b ^{9}}{6} + \frac{27 π a ^{6} b ^{12}}{6}

\frac{8 π a ^{12} b ^{3}}{6} = \frac{4 π a ^{12} b ^{3}}{3}

\frac{36 π a ^{10} b ^{6}}{6} = 6 π a^{10} b^{6}

\frac{54 π a ^{8} b ^{9}}{6} = 9 π a^{8} b^{9}

\frac{27 π a ^{6} b ^{12}}{6} = \frac{9 π a ^{6} b ^{12}}{2}

= - \frac{4 π a ^{12} b ^{3}}{3} + 6 π a^{10} b^{6} - 9 π a^{8} b^{9} + \frac{9 π a ^{6} b ^{12}}{2}

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