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Popular Algebra >

expand-(e^{-(pijn)/2}(2pijn+1))/(pi^2j^2n^2)

Solution

expand

- \frac{e ^{- \frac{πjn}{2}} ( 2 πjn + 1 )}{π ^{2} j ^{2} n ^{2}}

Solution

- \frac{2 e ^{- \frac{πjn}{2}}}{πjn} - \frac{e ^{- \frac{πjn}{2}}}{π ^{2} j ^{2} n ^{2}}

Solution steps

- \frac{e ^{- \frac{πjn}{2}} ( 2 πjn + 1 )}{π ^{2} j ^{2} n ^{2}}

Expand

e^{- \frac{πjn}{2}} (2 πjn + 1) : 2 πjn e^{- \frac{πjn}{2}} + e^{- \frac{πjn}{2}}

= - \frac{2 πjn e ^{- \frac{πjn}{2}} + e ^{- \frac{πjn}{2}}}{π ^{2} j ^{2} n ^{2}}

Apply the fraction rule:

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

\frac{2 πjn e ^{- \frac{πjn}{2}} + e ^{- \frac{πjn}{2}}}{π ^{2} j ^{2} n ^{2}} = - (\frac{2 πjn e ^{- \frac{πjn}{2}}}{π ^{2} j ^{2} n ^{2}}) - (\frac{e ^{- \frac{πjn}{2}}}{π ^{2} j ^{2} n ^{2}})

= - (\frac{2 πjn e ^{- \frac{πjn}{2}}}{π ^{2} j ^{2} n ^{2}}) - (\frac{e ^{- \frac{πjn}{2}}}{π ^{2} j ^{2} n ^{2}})

Remove parentheses:

(a) = a

= - \frac{2 πjn e ^{- \frac{πjn}{2}}}{π ^{2} j ^{2} n ^{2}} - \frac{e ^{- \frac{πjn}{2}}}{π ^{2} j ^{2} n ^{2}}

Cancel

\frac{2 πjn e ^{- \frac{πjn}{2}}}{π ^{2} j ^{2} n ^{2}} : \frac{2 e ^{- \frac{πjn}{2}}}{πjn}

= - \frac{2 e ^{- \frac{πjn}{2}}}{πjn} - \frac{e ^{- \frac{πjn}{2}}}{π ^{2} j ^{2} n ^{2}}

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