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인기 있는 대수학 >

확장 (x+1/(2sqrt(x)))^{14}

해법

확장

(x + \frac{1}{2 x})^{14}

해법

x^{14} + 7 x^{25} + \frac{91 x ^{11}}{4} + \frac{91 x ^{19}}{2} + \frac{1001 x ^{8}}{16} + \frac{1001 x ^{13}}{16} + \frac{3003 x ^{5}}{64} + \frac{429 x ^{7}}{16} + \frac{3003 x ^{2}}{256} + \frac{1001 x}{256} + \frac{1001}{1024 x} + \frac{91}{512 x ^{5}} + \frac{91}{4096 x ^{4}} + \frac{7}{4096 x ^{11}} + \frac{1}{16384 x ^{7}}

솔루션 단계

(x + \frac{1}{2 x})^{14}

이항 정리 적용:

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

a = x, b = \frac{1}{2 x}

= i = 0 \sum 14 (i 14) x^{(14 - i)} (\frac{1}{2 x})^{i}

합계 확장

= \frac{14 !}{0 ! ( 14 - 0 ) !} x^{14} (\frac{1}{2 x})^{0} + \frac{14 !}{1 ! ( 14 - 1 ) !} x^{13} (\frac{1}{2 x})^{1} + \frac{14 !}{2 ! ( 14 - 2 ) !} x^{12} (\frac{1}{2 x})^{2} + \frac{14 !}{3 ! ( 14 - 3 ) !} x^{11} (\frac{1}{2 x})^{3} + \frac{14 !}{4 ! ( 14 - 4 ) !} x^{10} (\frac{1}{2 x})^{4} + \frac{14 !}{5 ! ( 14 - 5 ) !} x^{9} (\frac{1}{2 x})^{5} + \frac{14 !}{6 ! ( 14 - 6 ) !} x^{8} (\frac{1}{2 x})^{6} + \frac{14 !}{7 ! ( 14 - 7 ) !} x^{7} (\frac{1}{2 x})^{7} + \frac{14 !}{8 ! ( 14 - 8 ) !} x^{6} (\frac{1}{2 x})^{8} + \frac{14 !}{9 ! ( 14 - 9 ) !} x^{5} (\frac{1}{2 x})^{9} + \frac{14 !}{10 ! ( 14 - 10 ) !} x^{4} (\frac{1}{2 x})^{10} + \frac{14 !}{11 ! ( 14 - 11 ) !} x^{3} (\frac{1}{2 x})^{11} + \frac{14 !}{12 ! ( 14 - 12 ) !} x^{2} (\frac{1}{2 x})^{12} + \frac{14 !}{13 ! ( 14 - 13 ) !} x^{1} (\frac{1}{2 x})^{13} + \frac{14 !}{14 ! ( 14 - 14 ) !} x^{0} (\frac{1}{2 x})^{14}

\frac{14 !}{0 ! ( 14 - 0 ) !} x^{14} (\frac{1}{2 x})^{0}

간소화하다 :

x^{14}

\frac{14 !}{1 ! ( 14 - 1 ) !} x^{13} (\frac{1}{2 x})^{1}

간소화하다 :

7 x^{\frac{25}{2}}

\frac{14 !}{2 ! ( 14 - 2 ) !} x^{12} (\frac{1}{2 x})^{2}

간소화하다 :

\frac{91 x ^{11}}{4}

\frac{14 !}{3 ! ( 14 - 3 ) !} x^{11} (\frac{1}{2 x})^{3}

간소화하다 :

\frac{91 x ^{11}}{2 ( x ) ^{3}}

\frac{14 !}{4 ! ( 14 - 4 ) !} x^{10} (\frac{1}{2 x})^{4}

간소화하다 :

\frac{1001 x ^{8}}{16}

\frac{14 !}{5 ! ( 14 - 5 ) !} x^{9} (\frac{1}{2 x})^{5}

간소화하다 :

\frac{1001 x ^{9}}{16 ( x ) ^{5}}

\frac{14 !}{6 ! ( 14 - 6 ) !} x^{8} (\frac{1}{2 x})^{6}

간소화하다 :

\frac{3003 x ^{5}}{64}

\frac{14 !}{7 ! ( 14 - 7 ) !} x^{7} (\frac{1}{2 x})^{7}

간소화하다 :

\frac{429 x ^{7}}{16 ( x ) ^{7}}

\frac{14 !}{8 ! ( 14 - 8 ) !} x^{6} (\frac{1}{2 x})^{8}

간소화하다 :

\frac{3003 x ^{2}}{256}

\frac{14 !}{9 ! ( 14 - 9 ) !} x^{5} (\frac{1}{2 x})^{9}

간소화하다 :

\frac{1001 x ^{5}}{256 ( x ) ^{9}}

\frac{14 !}{10 ! ( 14 - 10 ) !} x^{4} (\frac{1}{2 x})^{10}

간소화하다 :

\frac{1001}{1024 x}

\frac{14 !}{11 ! ( 14 - 11 ) !} x^{3} (\frac{1}{2 x})^{11}

간소화하다 :

\frac{91 x ^{3}}{512 ( x ) ^{11}}

\frac{14 !}{12 ! ( 14 - 12 ) !} x^{2} (\frac{1}{2 x})^{12}

간소화하다 :

\frac{91}{4096 x ^{4}}

\frac{14 !}{13 ! ( 14 - 13 ) !} x^{1} (\frac{1}{2 x})^{13}

간소화하다 :

\frac{7 x}{4096 ( x ) ^{13}}

\frac{14 !}{14 ! ( 14 - 14 ) !} x^{0} (\frac{1}{2 x})^{14}

간소화하다 :

\frac{1}{16384 x ^{7}}

= x^{14} + 7 x^{\frac{25}{2}} + \frac{91 x ^{11}}{4} + \frac{91 x ^{11}}{2 ( x ) ^{3}} + \frac{1001 x ^{8}}{16} + \frac{1001 x ^{9}}{16 ( x ) ^{5}} + \frac{3003 x ^{5}}{64} + \frac{429 x ^{7}}{16 ( x ) ^{7}} + \frac{3003 x ^{2}}{256} + \frac{1001 x ^{5}}{256 ( x ) ^{9}} + \frac{1001}{1024 x} + \frac{91 x ^{3}}{512 ( x ) ^{11}} + \frac{91}{4096 x ^{4}} + \frac{7 x}{4096 ( x ) ^{13}} + \frac{1}{16384 x ^{7}}

x^{14} + 7 x^{\frac{25}{2}} + \frac{91 x ^{11}}{4} + \frac{91 x ^{11}}{2 ( x ) ^{3}} + \frac{1001 x ^{8}}{16} + \frac{1001 x ^{9}}{16 ( x ) ^{5}} + \frac{3003 x ^{5}}{64} + \frac{429 x ^{7}}{16 ( x ) ^{7}} + \frac{3003 x ^{2}}{256} + \frac{1001 x ^{5}}{256 ( x ) ^{9}} + \frac{1001}{1024 x} + \frac{91 x ^{3}}{512 ( x ) ^{11}} + \frac{91}{4096 x ^{4}} + \frac{7 x}{4096 ( x ) ^{13}} + \frac{1}{16384 x ^{7}}

간소화하다 :

x^{14} + 7 x^{\frac{25}{2}} + \frac{91 x ^{11}}{4} + \frac{91 x ^{\frac{19}{2}}}{2} + \frac{1001 x ^{8}}{16} + \frac{1001 x ^{\frac{13}{2}}}{16} + \frac{3003 x ^{5}}{64} + \frac{429 x ^{\frac{7}{2}}}{16} + \frac{3003 x ^{2}}{256} + \frac{1001 x}{256} + \frac{1001}{1024 x} + \frac{91}{512 x ^{\frac{5}{2}}} + \frac{91}{4096 x ^{4}} + \frac{7}{4096 x ^{\frac{11}{2}}} + \frac{1}{16384 x ^{7}}

= x^{14} + 7 x^{\frac{25}{2}} + \frac{91 x ^{11}}{4} + \frac{91 x ^{\frac{19}{2}}}{2} + \frac{1001 x ^{8}}{16} + \frac{1001 x ^{\frac{13}{2}}}{16} + \frac{3003 x ^{5}}{64} + \frac{429 x ^{\frac{7}{2}}}{16} + \frac{3003 x ^{2}}{256} + \frac{1001 x}{256} + \frac{1001}{1024 x} + \frac{91}{512 x ^{\frac{5}{2}}} + \frac{91}{4096 x ^{4}} + \frac{7}{4096 x ^{\frac{11}{2}}} + \frac{1}{16384 x ^{7}}

x^{14} + 7 x^{\frac{25}{2}} + \frac{91 x ^{11}}{4} + \frac{91 x ^{\frac{19}{2}}}{2} + \frac{1001 x ^{8}}{16} + \frac{1001 x ^{\frac{13}{2}}}{16} + \frac{3003 x ^{5}}{64} + \frac{429 x ^{\frac{7}{2}}}{16} + \frac{3003 x ^{2}}{256} + \frac{1001 x}{256} + \frac{1001}{1024 x} + \frac{91}{512 x ^{\frac{5}{2}}} + \frac{91}{4096 x ^{4}} + \frac{7}{4096 x ^{\frac{11}{2}}} + \frac{1}{16384 x ^{7}}

간소화하다 :

x^{14} + 7 x^{25} + \frac{91 x ^{11}}{4} + \frac{91 x ^{19}}{2} + \frac{1001 x ^{8}}{16} + \frac{1001 x ^{13}}{16} + \frac{3003 x ^{5}}{64} + \frac{429 x ^{7}}{16} + \frac{3003 x ^{2}}{256} + \frac{1001 x}{256} + \frac{1001}{1024 x} + \frac{91}{512 x ^{5}} + \frac{91}{4096 x ^{4}} + \frac{7}{4096 x ^{11}} + \frac{1}{16384 x ^{7}}

= x^{14} + 7 x^{25} + \frac{91 x ^{11}}{4} + \frac{91 x ^{19}}{2} + \frac{1001 x ^{8}}{16} + \frac{1001 x ^{13}}{16} + \frac{3003 x ^{5}}{64} + \frac{429 x ^{7}}{16} + \frac{3003 x ^{2}}{256} + \frac{1001 x}{256} + \frac{1001}{1024 x} + \frac{91}{512 x ^{5}} + \frac{91}{4096 x ^{4}} + \frac{7}{4096 x ^{11}} + \frac{1}{16384 x ^{7}}

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