expandir (81(n^2+n)^2)/(4n^4)

Solução

expandir

\frac{81 ( n ^{2} + n ) ^{2}}{4 n ^{4}}

\frac{81}{4} + \frac{81}{2 n} + \frac{81}{4 n ^{2}}

Passos da solução

\frac{81 ( n ^{2} + n ) ^{2}}{4 n ^{4}}

(n^{2} + n)^{2} = n^{4} + 2 n^{3} + n^{2}

= \frac{81 ( n ^{4} + 2 n ^{3} + n ^{2} )}{4 n ^{4}}

Fatorar

n^{4} + 2 n^{3} + n^{2} : n^{2} (n^{2} + 2 n + 1)

= \frac{81 n ^{2} ( n ^{2} + 2 n + 1 )}{4 n ^{4}}

Cancelar

\frac{81 n ^{2} ( n ^{2} + 2 n + 1 )}{4 n ^{4}} : \frac{81 ( n ^{2} + 2 n + 1 )}{4 n ^{2}}

= \frac{81 ( n ^{2} + 2 n + 1 )}{4 n ^{2}}

Expandir

81 (n^{2} + 2 n + 1) : 81 n^{2} + 162 n + 81

= \frac{81 n ^{2} + 162 n + 81}{4 n ^{2}}

Aplicar as propriedades das frações:

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

\frac{81 n ^{2} + 162 n + 81}{4 n ^{2}} = \frac{81 n ^{2}}{4 n ^{2}} + \frac{162 n}{4 n ^{2}} + \frac{81}{4 n ^{2}}

= \frac{81 n ^{2}}{4 n ^{2}} + \frac{162 n}{4 n ^{2}} + \frac{81}{4 n ^{2}}

Cancelar

\frac{81 n ^{2}}{4 n ^{2}} : \frac{81}{4}

= \frac{81}{4} + \frac{162 n}{4 n ^{2}} + \frac{81}{4 n ^{2}}

Cancelar

\frac{162 n}{4 n ^{2}} : \frac{81}{2 n}

= \frac{81}{4} + \frac{81}{2 n} + \frac{81}{4 n ^{2}}

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