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Beliebt Algebra >

erweitern ((2x+(1))/((4x)))^{(20)}

Lösung

erweitern

(\frac{2 x + ( 1 )}{( 4 x )})^{(20)}

Lösung

\frac{1}{1048576} + \frac{5}{524288 x} + \frac{95}{2097152 x ^{2}} + \frac{149422080}{4 ^{20} x ^{3}} + \frac{317521920}{4 ^{20} x ^{4}} + \frac{508035072}{4 ^{20} x ^{5}} + \frac{635043840}{4 ^{20} x ^{6}} + \frac{635043840}{4 ^{20} x ^{7}} + \frac{515973120}{4 ^{20} x ^{8}} + \frac{343982080}{4 ^{20} x ^{9}} + \frac{189190144}{4 ^{20} x ^{10}} + \frac{20995}{268435456 x ^{11}} + \frac{62985}{2 ^{31} x ^{12}} + \frac{4845}{536870912 x ^{13}} + \frac{4845}{2 ^{31} x ^{14}} + \frac{969}{2 ^{31} x ^{15}} + \frac{4845}{2 ^{36} x ^{16}} + \frac{285}{2 ^{35} x ^{17}} + \frac{95}{2 ^{37} x ^{18}} + \frac{5}{2 ^{37} x ^{19}} + \frac{1}{4 ^{20} x ^{20}}

Schritte zur Lösung

(\frac{2 x + ( 1 )}{4 x})^{20}

Entferne die Klammern:

(a) = a

= (\frac{2 x + 1}{4 x})^{20}

Wende Exponentenregel an:

(\frac{a}{b})^{c} = \frac{a ^{c}}{b ^{c}}

= \frac{( 2 x + 1 ) ^{20}}{( 4 x ) ^{20}}

Wende Exponentenregel an:

(a \cdot b)^{n} = a^{n} b^{n}

(4 x)^{20} = 4^{20} x^{20}

= \frac{( 2 x + 1 ) ^{20}}{4 ^{20} x ^{20}}

(2 x + 1)^{20} = 1048576 x^{20} + 10485760 x^{19} + 49807360 x^{18} + 149422080 x^{17} + 317521920 x^{16} + 508035072 x^{15} + 635043840 x^{14} + 635043840 x^{13} + 515973120 x^{12} + 343982080 x^{11} + 189190144 x^{10} + 85995520 x^{9} + 32248320 x^{8} + 9922560 x^{7} + 2480640 x^{6} + 496128 x^{5} + 77520 x^{4} + 9120 x^{3} + 760 x^{2} + 40 x + 1

= \frac{1048576 x ^{20} + 10485760 x ^{19} + 49807360 x ^{18} + 149422080 x ^{17} + 317521920 x ^{16} + 508035072 x ^{15} + 635043840 x ^{14} + 635043840 x ^{13} + 515973120 x ^{12} + 343982080 x ^{11} + 189190144 x ^{10} + 85995520 x ^{9} + 32248320 x ^{8} + 9922560 x ^{7} + 2480640 x ^{6} + 496128 x ^{5} + 77520 x ^{4} + 9120 x ^{3} + 760 x ^{2} + 40 x + 1}{4 ^{20} x ^{20}}

Wende Bruchregel an:

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

\frac{1048576 x ^{20} + 10485760 x ^{19} + 49807360 x ^{18} + 149422080 x ^{17} + 317521920 x ^{16} + 508035072 x ^{15} + 635043840 x ^{14} + 635043840 x ^{13} + 515973120 x ^{12} + 343982080 x ^{11} + 189190144 x ^{10} + 85995520 x ^{9} + 32248320 x ^{8} + 9922560 x ^{7} + 2480640 x ^{6} + 496128 x ^{5} + 77520 x ^{4} + 9120 x ^{3} + 760 x ^{2} + 40 x + 1}{4 ^{20} x ^{20}} = \frac{1048576 x ^{20}}{4 ^{20} x ^{20}} + \frac{10485760 x ^{19}}{4 ^{20} x ^{20}} + \frac{49807360 x ^{18}}{4 ^{20} x ^{20}} + \frac{149422080 x ^{17}}{4 ^{20} x ^{20}} + \frac{317521920 x ^{16}}{4 ^{20} x ^{20}} + \frac{508035072 x ^{15}}{4 ^{20} x ^{20}} + \frac{635043840 x ^{14}}{4 ^{20} x ^{20}} + \frac{635043840 x ^{13}}{4 ^{20} x ^{20}} + \frac{515973120 x ^{12}}{4 ^{20} x ^{20}} + \frac{343982080 x ^{11}}{4 ^{20} x ^{20}} + \frac{189190144 x ^{10}}{4 ^{20} x ^{20}} + \frac{85995520 x ^{9}}{4 ^{20} x ^{20}} + \frac{32248320 x ^{8}}{4 ^{20} x ^{20}} + \frac{9922560 x ^{7}}{4 ^{20} x ^{20}} + \frac{2480640 x ^{6}}{4 ^{20} x ^{20}} + \frac{496128 x ^{5}}{4 ^{20} x ^{20}} + \frac{77520 x ^{4}}{4 ^{20} x ^{20}} + \frac{9120 x ^{3}}{4 ^{20} x ^{20}} + \frac{760 x ^{2}}{4 ^{20} x ^{20}} + \frac{40 x}{4 ^{20} x ^{20}} + \frac{1}{4 ^{20} x ^{20}}

= \frac{1048576 x ^{20}}{4 ^{20} x ^{20}} + \frac{10485760 x ^{19}}{4 ^{20} x ^{20}} + \frac{49807360 x ^{18}}{4 ^{20} x ^{20}} + \frac{149422080 x ^{17}}{4 ^{20} x ^{20}} + \frac{317521920 x ^{16}}{4 ^{20} x ^{20}} + \frac{508035072 x ^{15}}{4 ^{20} x ^{20}} + \frac{635043840 x ^{14}}{4 ^{20} x ^{20}} + \frac{635043840 x ^{13}}{4 ^{20} x ^{20}} + \frac{515973120 x ^{12}}{4 ^{20} x ^{20}} + \frac{343982080 x ^{11}}{4 ^{20} x ^{20}} + \frac{189190144 x ^{10}}{4 ^{20} x ^{20}} + \frac{85995520 x ^{9}}{4 ^{20} x ^{20}} + \frac{32248320 x ^{8}}{4 ^{20} x ^{20}} + \frac{9922560 x ^{7}}{4 ^{20} x ^{20}} + \frac{2480640 x ^{6}}{4 ^{20} x ^{20}} + \frac{496128 x ^{5}}{4 ^{20} x ^{20}} + \frac{77520 x ^{4}}{4 ^{20} x ^{20}} + \frac{9120 x ^{3}}{4 ^{20} x ^{20}} + \frac{760 x ^{2}}{4 ^{20} x ^{20}} + \frac{40 x}{4 ^{20} x ^{20}} + \frac{1}{4 ^{20} x ^{20}}

\frac{1048576 x ^{20}}{4 ^{20} x ^{20}} = \frac{1}{1048576}

\frac{10485760 x ^{19}}{4 ^{20} x ^{20}} = \frac{5}{524288 x}

\frac{49807360 x ^{18}}{4 ^{20} x ^{20}} = \frac{95}{2097152 x ^{2}}

\frac{149422080 x ^{17}}{4 ^{20} x ^{20}} = \frac{149422080}{4 ^{20} x ^{3}}

\frac{317521920 x ^{16}}{4 ^{20} x ^{20}} = \frac{317521920}{4 ^{20} x ^{4}}

\frac{508035072 x ^{15}}{4 ^{20} x ^{20}} = \frac{508035072}{4 ^{20} x ^{5}}

\frac{635043840 x ^{14}}{4 ^{20} x ^{20}} = \frac{635043840}{4 ^{20} x ^{6}}

\frac{635043840 x ^{13}}{4 ^{20} x ^{20}} = \frac{635043840}{4 ^{20} x ^{7}}

\frac{515973120 x ^{12}}{4 ^{20} x ^{20}} = \frac{515973120}{4 ^{20} x ^{8}}

\frac{343982080 x ^{11}}{4 ^{20} x ^{20}} = \frac{343982080}{4 ^{20} x ^{9}}

\frac{189190144 x ^{10}}{4 ^{20} x ^{20}} = \frac{189190144}{4 ^{20} x ^{10}}

\frac{85995520 x ^{9}}{4 ^{20} x ^{20}} = \frac{20995}{268435456 x ^{11}}

\frac{32248320 x ^{8}}{4 ^{20} x ^{20}} = \frac{62985}{2 ^{31} x ^{12}}

\frac{9922560 x ^{7}}{4 ^{20} x ^{20}} = \frac{4845}{536870912 x ^{13}}

\frac{2480640 x ^{6}}{4 ^{20} x ^{20}} = \frac{4845}{2 ^{31} x ^{14}}

\frac{496128 x ^{5}}{4 ^{20} x ^{20}} = \frac{969}{2 ^{31} x ^{15}}

\frac{77520 x ^{4}}{4 ^{20} x ^{20}} = \frac{4845}{2 ^{36} x ^{16}}

\frac{9120 x ^{3}}{4 ^{20} x ^{20}} = \frac{285}{2 ^{35} x ^{17}}

\frac{760 x ^{2}}{4 ^{20} x ^{20}} = \frac{95}{2 ^{37} x ^{18}}

\frac{40 x}{4 ^{20} x ^{20}} = \frac{5}{2 ^{37} x ^{19}}

= \frac{1}{1048576} + \frac{5}{524288 x} + \frac{95}{2097152 x ^{2}} + \frac{149422080}{4 ^{20} x ^{3}} + \frac{317521920}{4 ^{20} x ^{4}} + \frac{508035072}{4 ^{20} x ^{5}} + \frac{635043840}{4 ^{20} x ^{6}} + \frac{635043840}{4 ^{20} x ^{7}} + \frac{515973120}{4 ^{20} x ^{8}} + \frac{343982080}{4 ^{20} x ^{9}} + \frac{189190144}{4 ^{20} x ^{10}} + \frac{20995}{268435456 x ^{11}} + \frac{62985}{2 ^{31} x ^{12}} + \frac{4845}{536870912 x ^{13}} + \frac{4845}{2 ^{31} x ^{14}} + \frac{969}{2 ^{31} x ^{15}} + \frac{4845}{2 ^{36} x ^{16}} + \frac{285}{2 ^{35} x ^{17}} + \frac{95}{2 ^{37} x ^{18}} + \frac{5}{2 ^{37} x ^{19}} + \frac{1}{4 ^{20} x ^{20}}

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