expand ((3x^3+1)^6)/(x^{12)}

פתרון

expand

\frac{( 3 x ^{3} + 1 ) ^{6}}{x ^{12}}

729 x^{6} + 1458 x^{3} + 1215 + \frac{540}{x ^{3}} + \frac{135}{x ^{6}} + \frac{18}{x ^{9}} + \frac{1}{x ^{12}}

צעדי פתרון

\frac{( 3 x ^{3} + 1 ) ^{6}}{x ^{12}}

(3 x^{3} + 1)^{6} = 729 x^{18} + 1458 x^{15} + 1215 x^{12} + 540 x^{9} + 135 x^{6} + 18 x^{3} + 1

= \frac{729 x ^{18} + 1458 x ^{15} + 1215 x ^{12} + 540 x ^{9} + 135 x ^{6} + 18 x ^{3} + 1}{x ^{12}}

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

: השתמש בתכונת השברים הבאה

\frac{729 x ^{18} + 1458 x ^{15} + 1215 x ^{12} + 540 x ^{9} + 135 x ^{6} + 18 x ^{3} + 1}{x ^{12}} = \frac{729 x ^{18}}{x ^{12}} + \frac{1458 x ^{15}}{x ^{12}} + \frac{1215 x ^{12}}{x ^{12}} + \frac{540 x ^{9}}{x ^{12}} + \frac{135 x ^{6}}{x ^{12}} + \frac{18 x ^{3}}{x ^{12}} + \frac{1}{x ^{12}}

= \frac{729 x ^{18}}{x ^{12}} + \frac{1458 x ^{15}}{x ^{12}} + \frac{1215 x ^{12}}{x ^{12}} + \frac{540 x ^{9}}{x ^{12}} + \frac{135 x ^{6}}{x ^{12}} + \frac{18 x ^{3}}{x ^{12}} + \frac{1}{x ^{12}}

\frac{729 x ^{18}}{x ^{12}} = 729 x^{6}

\frac{1458 x ^{15}}{x ^{12}} = 1458 x^{3}

\frac{1215 x ^{12}}{x ^{12}} = 1215

\frac{540 x ^{9}}{x ^{12}} = \frac{540}{x ^{3}}

\frac{135 x ^{6}}{x ^{12}} = \frac{135}{x ^{6}}

\frac{18 x ^{3}}{x ^{12}} = \frac{18}{x ^{9}}

= 729 x^{6} + 1458 x^{3} + 1215 + \frac{540}{x ^{3}} + \frac{135}{x ^{6}} + \frac{18}{x ^{9}} + \frac{1}{x ^{12}}

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