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인기 있는 대수학 >

확장 (1/2 sqrt(a)+2/3 a^{4/5})^{12}

해법

확장

(\frac{1}{2} a + \frac{2}{3} a^{\frac{4}{5}})^{12}

해법

\frac{a ^{6}}{4096} + \frac{a ^{\frac{63}{10}}}{256} + \frac{11 a ^{\frac{33}{5}}}{384} + \frac{55 a ^{\frac{69}{10}}}{432} + \frac{55 a ^{\frac{36}{5}}}{144} + \frac{22 a ^{\frac{15}{2}}}{27} + \frac{308 a ^{\frac{39}{5}} + 352 a ^{\frac{81}{10}}}{243} + \frac{880 a ^{\frac{42}{5}}}{729} + \frac{14080 a ^{\frac{87}{10}} + 5632 a ^{9}}{19683} + \frac{4096 a ^{\frac{93}{10}}}{59049} + \frac{4096 a ^{\frac{48}{5}}}{531441}

솔루션 단계

(\frac{1}{2} a + \frac{2}{3} a^{\frac{4}{5}})^{12}

이항 정리 적용:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = \frac{1}{2} a, b = \frac{2}{3} a^{\frac{4}{5}}

= i = 0 \sum 12 (i 12) (\frac{1}{2} a)^{(12 - i)} (\frac{2}{3} a^{\frac{4}{5}})^{i}

합계 확장

= \frac{12 !}{0 ! ( 12 - 0 ) !} (\frac{1}{2} a)^{12} (\frac{2}{3} a^{\frac{4}{5}})^{0} + \frac{12 !}{1 ! ( 12 - 1 ) !} (\frac{1}{2} a)^{11} (\frac{2}{3} a^{\frac{4}{5}})^{1} + \frac{12 !}{2 ! ( 12 - 2 ) !} (\frac{1}{2} a)^{10} (\frac{2}{3} a^{\frac{4}{5}})^{2} + \frac{12 !}{3 ! ( 12 - 3 ) !} (\frac{1}{2} a)^{9} (\frac{2}{3} a^{\frac{4}{5}})^{3} + \frac{12 !}{4 ! ( 12 - 4 ) !} (\frac{1}{2} a)^{8} (\frac{2}{3} a^{\frac{4}{5}})^{4} + \frac{12 !}{5 ! ( 12 - 5 ) !} (\frac{1}{2} a)^{7} (\frac{2}{3} a^{\frac{4}{5}})^{5} + \frac{12 !}{6 ! ( 12 - 6 ) !} (\frac{1}{2} a)^{6} (\frac{2}{3} a^{\frac{4}{5}})^{6} + \frac{12 !}{7 ! ( 12 - 7 ) !} (\frac{1}{2} a)^{5} (\frac{2}{3} a^{\frac{4}{5}})^{7} + \frac{12 !}{8 ! ( 12 - 8 ) !} (\frac{1}{2} a)^{4} (\frac{2}{3} a^{\frac{4}{5}})^{8} + \frac{12 !}{9 ! ( 12 - 9 ) !} (\frac{1}{2} a)^{3} (\frac{2}{3} a^{\frac{4}{5}})^{9} + \frac{12 !}{10 ! ( 12 - 10 ) !} (\frac{1}{2} a)^{2} (\frac{2}{3} a^{\frac{4}{5}})^{10} + \frac{12 !}{11 ! ( 12 - 11 ) !} (\frac{1}{2} a)^{1} (\frac{2}{3} a^{\frac{4}{5}})^{11} + \frac{12 !}{12 ! ( 12 - 12 ) !} (\frac{1}{2} a)^{0} (\frac{2}{3} a^{\frac{4}{5}})^{12}

\frac{12 !}{0 ! ( 12 - 0 ) !} (\frac{1}{2} a)^{12} (\frac{2}{3} a^{\frac{4}{5}})^{0}

간소화하다 :

\frac{a ^{6}}{4096}

\frac{12 !}{1 ! ( 12 - 1 ) !} (\frac{1}{2} a)^{11} (\frac{2}{3} a^{\frac{4}{5}})^{1}

간소화하다 :

\frac{a ^{\frac{4}{5}} ( a ) ^{11}}{256}

\frac{12 !}{2 ! ( 12 - 2 ) !} (\frac{1}{2} a)^{10} (\frac{2}{3} a^{\frac{4}{5}})^{2}

간소화하다 :

\frac{11 a ^{\frac{33}{5}}}{384}

\frac{12 !}{3 ! ( 12 - 3 ) !} (\frac{1}{2} a)^{9} (\frac{2}{3} a^{\frac{4}{5}})^{3}

간소화하다 :

\frac{55 a ^{\frac{69}{10}}}{432}

\frac{12 !}{4 ! ( 12 - 4 ) !} (\frac{1}{2} a)^{8} (\frac{2}{3} a^{\frac{4}{5}})^{4}

간소화하다 :

\frac{55 a ^{\frac{36}{5}}}{144}

\frac{12 !}{5 ! ( 12 - 5 ) !} (\frac{1}{2} a)^{7} (\frac{2}{3} a^{\frac{4}{5}})^{5}

간소화하다 :

\frac{22 a ^{4} ( a ) ^{7}}{27}

\frac{12 !}{6 ! ( 12 - 6 ) !} (\frac{1}{2} a)^{6} (\frac{2}{3} a^{\frac{4}{5}})^{6}

간소화하다 :

\frac{308 a ^{\frac{39}{5}}}{243}

\frac{12 !}{7 ! ( 12 - 7 ) !} (\frac{1}{2} a)^{5} (\frac{2}{3} a^{\frac{4}{5}})^{7}

간소화하다 :

\frac{352 a ^{\frac{81}{10}}}{243}

\frac{12 !}{8 ! ( 12 - 8 ) !} (\frac{1}{2} a)^{4} (\frac{2}{3} a^{\frac{4}{5}})^{8}

간소화하다 :

\frac{880 a ^{\frac{42}{5}}}{729}

\frac{12 !}{9 ! ( 12 - 9 ) !} (\frac{1}{2} a)^{3} (\frac{2}{3} a^{\frac{4}{5}})^{9}

간소화하다 :

\frac{14080 a ^{\frac{87}{10}}}{19683}

\frac{12 !}{10 ! ( 12 - 10 ) !} (\frac{1}{2} a)^{2} (\frac{2}{3} a^{\frac{4}{5}})^{10}

간소화하다 :

\frac{5632 a ^{9}}{19683}

\frac{12 !}{11 ! ( 12 - 11 ) !} (\frac{1}{2} a)^{1} (\frac{2}{3} a^{\frac{4}{5}})^{11}

간소화하다 :

\frac{4096 a ^{\frac{93}{10}}}{59049}

\frac{12 !}{12 ! ( 12 - 12 ) !} (\frac{1}{2} a)^{0} (\frac{2}{3} a^{\frac{4}{5}})^{12}

간소화하다 :

\frac{4096 a ^{\frac{48}{5}}}{531441}

= \frac{a ^{6}}{4096} + \frac{a ^{\frac{4}{5}} ( a ) ^{11}}{256} + \frac{11 a ^{\frac{33}{5}}}{384} + \frac{55 a ^{\frac{69}{10}}}{432} + \frac{55 a ^{\frac{36}{5}}}{144} + \frac{22 a ^{4} ( a ) ^{7}}{27} + \frac{308 a ^{\frac{39}{5}}}{243} + \frac{352 a ^{\frac{81}{10}}}{243} + \frac{880 a ^{\frac{42}{5}}}{729} + \frac{14080 a ^{\frac{87}{10}}}{19683} + \frac{5632 a ^{9}}{19683} + \frac{4096 a ^{\frac{93}{10}}}{59049} + \frac{4096 a ^{\frac{48}{5}}}{531441}

\frac{a ^{6}}{4096} + \frac{a ^{\frac{4}{5}} ( a ) ^{11}}{256} + \frac{11 a ^{\frac{33}{5}}}{384} + \frac{55 a ^{\frac{69}{10}}}{432} + \frac{55 a ^{\frac{36}{5}}}{144} + \frac{22 a ^{4} ( a ) ^{7}}{27} + \frac{308 a ^{\frac{39}{5}}}{243} + \frac{352 a ^{\frac{81}{10}}}{243} + \frac{880 a ^{\frac{42}{5}}}{729} + \frac{14080 a ^{\frac{87}{10}}}{19683} + \frac{5632 a ^{9}}{19683} + \frac{4096 a ^{\frac{93}{10}}}{59049} + \frac{4096 a ^{\frac{48}{5}}}{531441}

간소화하다 :

\frac{a ^{6}}{4096} + \frac{a ^{\frac{63}{10}}}{256} + \frac{11 a ^{\frac{33}{5}}}{384} + \frac{55 a ^{\frac{69}{10}}}{432} + \frac{55 a ^{\frac{36}{5}}}{144} + \frac{22 a ^{\frac{15}{2}}}{27} + \frac{308 a ^{\frac{39}{5}} + 352 a ^{\frac{81}{10}}}{243} + \frac{880 a ^{\frac{42}{5}}}{729} + \frac{14080 a ^{\frac{87}{10}} + 5632 a ^{9}}{19683} + \frac{4096 a ^{\frac{93}{10}}}{59049} + \frac{4096 a ^{\frac{48}{5}}}{531441}

= \frac{a ^{6}}{4096} + \frac{a ^{\frac{63}{10}}}{256} + \frac{11 a ^{\frac{33}{5}}}{384} + \frac{55 a ^{\frac{69}{10}}}{432} + \frac{55 a ^{\frac{36}{5}}}{144} + \frac{22 a ^{\frac{15}{2}}}{27} + \frac{308 a ^{\frac{39}{5}} + 352 a ^{\frac{81}{10}}}{243} + \frac{880 a ^{\frac{42}{5}}}{729} + \frac{14080 a ^{\frac{87}{10}} + 5632 a ^{9}}{19683} + \frac{4096 a ^{\frac{93}{10}}}{59049} + \frac{4096 a ^{\frac{48}{5}}}{531441}

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