he

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

פּוֹפּוּלָרִי אלגברה >

expand (x-2/(sqrt(x)))^{15}

פתרון

expand

(x - \frac{2}{x})^{15}

פתרון

x^{15} - 30 x^{27} + 420 x^{12} - 3640 x^{21} - 96096 x^{15} - 823680 x^{9} - 2562560 x^{3} + 21840 x^{9} + 320320 x^{6} + 1647360 x^{3} + 3075072 - \frac{2795520}{x ^{3}} + \frac{1863680}{x ^{3}} - \frac{860160}{x ^{9}} + \frac{245760}{x ^{6}} - \frac{32768}{x ^{15}}

צעדי פתרון

(x - \frac{2}{x})^{15}

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

:הפעל את הבינום של ניוטון

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

= i = 0 \sum 15 (i 15) x^{(15 - i)} (- \frac{2}{x})^{i}

הרחב את הסכום

= \frac{15 !}{0 ! ( 15 - 0 ) !} x^{15} (- \frac{2}{x})^{0} + \frac{15 !}{1 ! ( 15 - 1 ) !} x^{14} (- \frac{2}{x})^{1} + \frac{15 !}{2 ! ( 15 - 2 ) !} x^{13} (- \frac{2}{x})^{2} + \frac{15 !}{3 ! ( 15 - 3 ) !} x^{12} (- \frac{2}{x})^{3} + \frac{15 !}{4 ! ( 15 - 4 ) !} x^{11} (- \frac{2}{x})^{4} + \frac{15 !}{5 ! ( 15 - 5 ) !} x^{10} (- \frac{2}{x})^{5} + \frac{15 !}{6 ! ( 15 - 6 ) !} x^{9} (- \frac{2}{x})^{6} + \frac{15 !}{7 ! ( 15 - 7 ) !} x^{8} (- \frac{2}{x})^{7} + \frac{15 !}{8 ! ( 15 - 8 ) !} x^{7} (- \frac{2}{x})^{8} + \frac{15 !}{9 ! ( 15 - 9 ) !} x^{6} (- \frac{2}{x})^{9} + \frac{15 !}{10 ! ( 15 - 10 ) !} x^{5} (- \frac{2}{x})^{10} + \frac{15 !}{11 ! ( 15 - 11 ) !} x^{4} (- \frac{2}{x})^{11} + \frac{15 !}{12 ! ( 15 - 12 ) !} x^{3} (- \frac{2}{x})^{12} + \frac{15 !}{13 ! ( 15 - 13 ) !} x^{2} (- \frac{2}{x})^{13} + \frac{15 !}{14 ! ( 15 - 14 ) !} x^{1} (- \frac{2}{x})^{14} + \frac{15 !}{15 ! ( 15 - 15 ) !} x^{0} (- \frac{2}{x})^{15}

\frac{15 !}{0 ! ( 15 - 0 ) !} x^{15} (- \frac{2}{x})^{0}

פשט את:

x^{15}

\frac{15 !}{1 ! ( 15 - 1 ) !} x^{14} (- \frac{2}{x})^{1}

פשט את:

- 30 x^{\frac{27}{2}}

\frac{15 !}{2 ! ( 15 - 2 ) !} x^{13} (- \frac{2}{x})^{2}

פשט את:

420 x^{12}

\frac{15 !}{3 ! ( 15 - 3 ) !} x^{12} (- \frac{2}{x})^{3}

פשט את:

- \frac{3640 x ^{12}}{( x ) ^{3}}

\frac{15 !}{4 ! ( 15 - 4 ) !} x^{11} (- \frac{2}{x})^{4}

פשט את:

21840 x^{9}

\frac{15 !}{5 ! ( 15 - 5 ) !} x^{10} (- \frac{2}{x})^{5}

פשט את:

- \frac{96096 x ^{10}}{( x ) ^{5}}

\frac{15 !}{6 ! ( 15 - 6 ) !} x^{9} (- \frac{2}{x})^{6}

פשט את:

320320 x^{6}

\frac{15 !}{7 ! ( 15 - 7 ) !} x^{8} (- \frac{2}{x})^{7}

פשט את:

- \frac{823680 x ^{8}}{( x ) ^{7}}

\frac{15 !}{8 ! ( 15 - 8 ) !} x^{7} (- \frac{2}{x})^{8}

פשט את:

1647360 x^{3}

\frac{15 !}{9 ! ( 15 - 9 ) !} x^{6} (- \frac{2}{x})^{9}

פשט את:

- \frac{2562560 x ^{6}}{( x ) ^{9}}

\frac{15 !}{10 ! ( 15 - 10 ) !} x^{5} (- \frac{2}{x})^{10}

פשט את:

3075072

\frac{15 !}{11 ! ( 15 - 11 ) !} x^{4} (- \frac{2}{x})^{11}

פשט את:

- \frac{2795520 x ^{4}}{( x ) ^{11}}

\frac{15 !}{12 ! ( 15 - 12 ) !} x^{3} (- \frac{2}{x})^{12}

פשט את:

\frac{1863680}{x ^{3}}

\frac{15 !}{13 ! ( 15 - 13 ) !} x^{2} (- \frac{2}{x})^{13}

פשט את:

- \frac{860160 x ^{2}}{( x ) ^{13}}

\frac{15 !}{14 ! ( 15 - 14 ) !} x^{1} (- \frac{2}{x})^{14}

פשט את:

\frac{245760}{x ^{6}}

\frac{15 !}{15 ! ( 15 - 15 ) !} x^{0} (- \frac{2}{x})^{15}

פשט את:

- \frac{32768}{( x ) ^{15}}

= x^{15} - 30 x^{\frac{27}{2}} + 420 x^{12} - \frac{3640 x ^{12}}{( x ) ^{3}} + 21840 x^{9} - \frac{96096 x ^{10}}{( x ) ^{5}} + 320320 x^{6} - \frac{823680 x ^{8}}{( x ) ^{7}} + 1647360 x^{3} - \frac{2562560 x ^{6}}{( x ) ^{9}} + 3075072 - \frac{2795520 x ^{4}}{( x ) ^{11}} + \frac{1863680}{x ^{3}} - \frac{860160 x ^{2}}{( x ) ^{13}} + \frac{245760}{x ^{6}} - \frac{32768}{( x ) ^{15}}

\frac{3640 x ^{12}}{( x ) ^{3}} = \frac{3640 x ^{11}}{x}

\frac{96096 x ^{10}}{( x ) ^{5}} = \frac{96096 x ^{8}}{x}

\frac{823680 x ^{8}}{( x ) ^{7}} = \frac{823680 x ^{5}}{x}

\frac{2562560 x ^{6}}{( x ) ^{9}} = \frac{2562560 x ^{2}}{x}

\frac{2795520 x ^{4}}{( x ) ^{11}} = \frac{2795520}{x ^{\frac{3}{2}}}

\frac{860160 x ^{2}}{( x ) ^{13}} = \frac{860160}{x ^{\frac{9}{2}}}

\frac{32768}{( x ) ^{15}} = \frac{32768}{x ^{\frac{15}{2}}}

= x^{15} - 30 x^{\frac{27}{2}} + 420 x^{12} - \frac{3640 x ^{11}}{x} + 21840 x^{9} - \frac{96096 x ^{8}}{x} + 320320 x^{6} - \frac{823680 x ^{5}}{x} + 1647360 x^{3} - \frac{2562560 x ^{2}}{x} + 3075072 - \frac{2795520}{x ^{\frac{3}{2}}} + \frac{1863680}{x ^{3}} - \frac{860160}{x ^{\frac{9}{2}}} + \frac{245760}{x ^{6}} - \frac{32768}{x ^{\frac{15}{2}}}

x^{15} - 30 x^{\frac{27}{2}} + 420 x^{12} - \frac{3640 x ^{11}}{x} + 21840 x^{9} - \frac{96096 x ^{8}}{x} + 320320 x^{6} - \frac{823680 x ^{5}}{x} + 1647360 x^{3} - \frac{2562560 x ^{2}}{x} + 3075072 - \frac{2795520}{x ^{\frac{3}{2}}} + \frac{1863680}{x ^{3}} - \frac{860160}{x ^{\frac{9}{2}}} + \frac{245760}{x ^{6}} - \frac{32768}{x ^{\frac{15}{2}}}

אחד את:

x^{15} - 30 x^{\frac{27}{2}} + 420 x^{12} - 104 x^{\frac{3}{2}} (35 x^{9} + 924 x^{6} + 7920 x^{3} + 24640) + 21840 x^{9} + 320320 x^{6} + 1647360 x^{3} + 3075072 - \frac{2795520}{x ^{\frac{3}{2}}} + \frac{1863680}{x ^{3}} - \frac{860160}{x ^{\frac{9}{2}}} + \frac{245760}{x ^{6}} - \frac{32768}{x ^{\frac{15}{2}}}

= x^{15} - 30 x^{\frac{27}{2}} + 420 x^{12} - 104 x^{\frac{3}{2}} (35 x^{9} + 924 x^{6} + 7920 x^{3} + 24640) + 21840 x^{9} + 320320 x^{6} + 1647360 x^{3} + 3075072 - \frac{2795520}{x ^{\frac{3}{2}}} + \frac{1863680}{x ^{3}} - \frac{860160}{x ^{\frac{9}{2}}} + \frac{245760}{x ^{6}} - \frac{32768}{x ^{\frac{15}{2}}}

- 104 x^{\frac{3}{2}} (35 x^{9} + 924 x^{6} + 7920 x^{3} + 24640)

הרחב את:

- 3640 x^{\frac{21}{2}} - 96096 x^{\frac{15}{2}} - 823680 x^{\frac{9}{2}} - 2562560 x^{\frac{3}{2}}

= x^{15} - 30 x^{\frac{27}{2}} + 420 x^{12} - 3640 x^{\frac{21}{2}} - 96096 x^{\frac{15}{2}} - 823680 x^{\frac{9}{2}} - 2562560 x^{\frac{3}{2}} + 21840 x^{9} + 320320 x^{6} + 1647360 x^{3} + 3075072 - \frac{2795520}{x ^{\frac{3}{2}}} + \frac{1863680}{x ^{3}} - \frac{860160}{x ^{\frac{9}{2}}} + \frac{245760}{x ^{6}} - \frac{32768}{x ^{\frac{15}{2}}}

x^{15} - 30 x^{\frac{27}{2}} + 420 x^{12} - 3640 x^{\frac{21}{2}} - 96096 x^{\frac{15}{2}} - 823680 x^{\frac{9}{2}} - 2562560 x^{\frac{3}{2}} + 21840 x^{9} + 320320 x^{6} + 1647360 x^{3} + 3075072 - \frac{2795520}{x ^{\frac{3}{2}}} + \frac{1863680}{x ^{3}} - \frac{860160}{x ^{\frac{9}{2}}} + \frac{245760}{x ^{6}} - \frac{32768}{x ^{\frac{15}{2}}}

פשט את:

x^{15} - 30 x^{27} + 420 x^{12} - 3640 x^{21} - 96096 x^{15} - 823680 x^{9} - 2562560 x^{3} + 21840 x^{9} + 320320 x^{6} + 1647360 x^{3} + 3075072 - \frac{2795520}{x ^{3}} + \frac{1863680}{x ^{3}} - \frac{860160}{x ^{9}} + \frac{245760}{x ^{6}} - \frac{32768}{x ^{15}}

= x^{15} - 30 x^{27} + 420 x^{12} - 3640 x^{21} - 96096 x^{15} - 823680 x^{9} - 2562560 x^{3} + 21840 x^{9} + 320320 x^{6} + 1647360 x^{3} + 3075072 - \frac{2795520}{x ^{3}} + \frac{1863680}{x ^{3}} - \frac{860160}{x ^{9}} + \frac{245760}{x ^{6}} - \frac{32768}{x ^{15}}

דוגמאות פופולריות

expand limx\Rightarrow 1(x^3+3x^2-x-3)(x^2-1)

e x p an d l im x \Rightarrow 1 (x^{3} + 3 x^{2} - x - 3) (x^{2} - 1)

simplify (z-1/2)(z-1/3)

s im pl i f y (z - \frac{1}{2}) (z - \frac{1}{3})

expand sin(pi(x+1))

e x p an d sin (π (x + 1))

expand (1-e^{-x})(1-e^{-y})

e x p an d (1 - e^{- x}) (1 - e^{- y})

expand (y^5-(x^2y^3))/((x^2+y^2)^2)

e x p an d \frac{y ^{5} - ( x ^{2} y ^{3} )}{( x ^{2} + y ^{2} ) ^{2}}