확장 ((25x^2+2x)^5)/((3sqrt(x)+8x^3)^3)

해법

확장

\frac{( 25 x ^{2} + 2 x ) ^{5}}{( 3 x + 8 x ^{3} ) ^{3}}

\frac{9765625 x ^{9} + 3906250 x ^{8} + 625000 x ^{7} + 50000 x ^{6} + 2000 x ^{5} + 32 x ^{4}}{576 x ^{5} x + 216 x ^{3} + 27 x + 512 x ^{8}}

솔루션 단계

\frac{( 25 x ^{2} + 2 x ) ^{5}}{( 3 x + 8 x ^{3} ) ^{3}}

(3 x + 8 x^{3})^{3} = 27 x^{\frac{3}{2}} + 216 x^{4} + 576 x^{6} x + 512 x^{9}

= \frac{( 25 x ^{2} + 2 x ) ^{5}}{27 x ^{\frac{3}{2}} + 216 x ^{4} + 576 x ^{6} x + 512 x ^{9}}

(25 x^{2} + 2 x)^{5} = 9765625 x^{10} + 3906250 x^{9} + 625000 x^{8} + 50000 x^{7} + 2000 x^{6} + 32 x^{5}

= \frac{9765625 x ^{10} + 3906250 x ^{9} + 625000 x ^{8} + 50000 x ^{7} + 2000 x ^{6} + 32 x ^{5}}{27 x ^{\frac{3}{2}} + 216 x ^{4} + 576 x ^{6} x + 512 x ^{9}}

9765625 x^{10} + 3906250 x^{9} + 625000 x^{8} + 50000 x^{7} + 2000 x^{6} + 32 x^{5}

요인:

x^{5} (9765625 x^{5} + 3906250 x^{4} + 625000 x^{3} + 50000 x^{2} + 2000 x + 32)

= \frac{x ^{5} ( 9765625 x ^{5} + 3906250 x ^{4} + 625000 x ^{3} + 50000 x ^{2} + 2000 x + 32 )}{27 x ^{\frac{3}{2}} + 216 x ^{4} + 576 x ^{6} x + 512 x ^{9}}

27 x^{\frac{3}{2}} + 216 x^{4} + 576 x^{6} x + 512 x^{9}

요인:

x (9 (x^{3} \cdot 24 + 3 x + x^{5} \cdot 64 x) + 512 x^{8})

= \frac{x ^{5} ( 9765625 x ^{5} + 3906250 x ^{4} + 625000 x ^{3} + 50000 x ^{2} + 2000 x + 32 )}{x ( 9 ( x ^{3} \cdot 24 + 3 x + x ^{5} \cdot 64 x ) + 512 x ^{8} )}

공통 요인 취소:

x

= \frac{x ^{4} ( 9765625 x ^{5} + 3906250 x ^{4} + 625000 x ^{3} + 50000 x ^{2} + 2000 x + 32 )}{9 ( 64 x ^{5} x + 24 x ^{3} + 3 x ) + 512 x ^{8}}

9 (64 x^{5} x + 24 x^{3} + 3 x) + 512 x^{8}

확대한다:

576 x^{5} x + 216 x^{3} + 27 x + 512 x^{8}

= \frac{x ^{4} ( 9765625 x ^{5} + 3906250 x ^{4} + 625000 x ^{3} + 50000 x ^{2} + 2000 x + 32 )}{576 x ^{5} x + 216 x ^{3} + 27 x + 512 x ^{8}}

x^{4} (9765625 x^{5} + 3906250 x^{4} + 625000 x^{3} + 50000 x^{2} + 2000 x + 32)

확대한다:

9765625 x^{9} + 3906250 x^{8} + 625000 x^{7} + 50000 x^{6} + 2000 x^{5} + 32 x^{4}

= \frac{9765625 x ^{9} + 3906250 x ^{8} + 625000 x ^{7} + 50000 x ^{6} + 2000 x ^{5} + 32 x ^{4}}{576 x ^{5} x + 216 x ^{3} + 27 x + 512 x ^{8}}