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Popular Algebra >

expand ((e^{2x}+6)^6)/(12)

Solution

expand

\frac{( e ^{2 x} + 6 ) ^{6}}{12}

Solution

\frac{e ^{12 x}}{12} + 3 e^{10 x} + 45 e^{8 x} + 360 e^{6 x} + 1620 e^{4 x} + 3888 e^{2 x} + 3888

Solution steps

\frac{( e ^{2 x} + 6 ) ^{6}}{12}

(e^{2 x} + 6)^{6} = e^{12 x} + 36 e^{10 x} + 540 e^{8 x} + 4320 e^{6 x} + 19440 e^{4 x} + 46656 e^{2 x} + 46656

= \frac{e ^{12 x} + 36 e ^{10 x} + 540 e ^{8 x} + 4320 e ^{6 x} + 19440 e ^{4 x} + 46656 e ^{2 x} + 46656}{12}

Apply the fraction rule:

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

\frac{e ^{12 x} + 36 e ^{10 x} + 540 e ^{8 x} + 4320 e ^{6 x} + 19440 e ^{4 x} + 46656 e ^{2 x} + 46656}{12} = \frac{e ^{12 x}}{12} + \frac{36 e ^{10 x}}{12} + \frac{540 e ^{8 x}}{12} + \frac{4320 e ^{6 x}}{12} + \frac{19440 e ^{4 x}}{12} + \frac{46656 e ^{2 x}}{12} + \frac{46656}{12}

= \frac{e ^{12 x}}{12} + \frac{36 e ^{10 x}}{12} + \frac{540 e ^{8 x}}{12} + \frac{4320 e ^{6 x}}{12} + \frac{19440 e ^{4 x}}{12} + \frac{46656 e ^{2 x}}{12} + \frac{46656}{12}

\frac{36 e ^{10 x}}{12} = 3 e^{10 x}

\frac{540 e ^{8 x}}{12} = 45 e^{8 x}

\frac{4320 e ^{6 x}}{12} = 360 e^{6 x}

\frac{19440 e ^{4 x}}{12} = 1620 e^{4 x}

\frac{46656 e ^{2 x}}{12} = 3888 e^{2 x}

\frac{46656}{12} = 3888

= \frac{e ^{12 x}}{12} + 3 e^{10 x} + 45 e^{8 x} + 360 e^{6 x} + 1620 e^{4 x} + 3888 e^{2 x} + 3888

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