What is expand (10^6K)/((is+10^2)^3) ?

The solution to expand (10^6K)/((is+10^2)^3) is

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expand (10^6K)/((is+10^2)^3)

expand

\frac{1 0 ^{6} K}{( i s + 1 0 ^{2} ) ^{3}}

\frac{1000000000000 K - 300000000 K s ^{2}}{100000 0 ^{2} + s ^{6} + 300000000 s ^{2} + 30000 s ^{4}} + i \frac{1000000 K s ^{3} - 30000000000 Ks}{100000 0 ^{2} + s ^{6} + 300000000 s ^{2} + 30000 s ^{4}}

Solution steps

\frac{1 0 ^{6} K}{( i s + 1 0 ^{2} ) ^{3}}

1 0^{2} = 100

= \frac{1 0 ^{6} K}{( i s + 100 ) ^{3}}

(i s + 100)^{3} = - i s^{3} - 300 s^{2} + 30000 i s + 1000000

= \frac{1 0 ^{6} K}{( - i s ^{3} - 300 s ^{2} + 30000 i s + 1000000 )}

Refine

= \frac{1000000 K}{- i s ^{3} - 300 s ^{2} + 30000 i s + 1000000}

Rationalize

\frac{1000000 K}{- i s ^{3} - 300 s ^{2} + 30000 i s + 1000000} : \frac{1000000 K ( - 300 s ^{2} + i ( s ^{3} - 30000 s ) + 1000000 )}{s ^{6} + 30000 s ^{4} + 300000000 s ^{2} + 100000 0 ^{2}}

= \frac{1000000 K ( - 300 s ^{2} + i ( s ^{3} - 30000 s ) + 1000000 )}{s ^{6} + 30000 s ^{4} + 300000000 s ^{2} + 100000 0 ^{2}}

Rewrite

\frac{1000000 K ( - 300 s ^{2} + 1000000 + i ( s ^{3} - 30000 s ) )}{s ^{6} + 30000 s ^{4} + 300000000 s ^{2} + 100000 0 ^{2}}

in standard complex form:

\frac{- 300000000 K s ^{2} + 1000000000000 K}{s ^{6} + 30000 s ^{4} + 300000000 s ^{2} + 100000 0 ^{2}} + \frac{1000000 K s ^{3} - 30000000000 Ks}{s ^{6} + 30000 s ^{4} + 300000000 s ^{2} + 100000 0 ^{2}} i

= \frac{- 300000000 K s ^{2} + 1000000000000 K}{s ^{6} + 30000 s ^{4} + 300000000 s ^{2} + 100000 0 ^{2}} + \frac{1000000 K s ^{3} - 30000000000 Ks}{s ^{6} + 30000 s ^{4} + 300000000 s ^{2} + 100000 0 ^{2}} i

\frac{- csc ^{2} ( x ) ( x + 1 ) - cot ( x )}{( x + 1 ) ^{2}}

sin (x) + y (cos (x)) - sin (x) (cos (x) + y)

\frac{4}{( 2 - t ) ^{3}}

(sin^{2} (t)) (5 - (t^{2}))

e^{e^{x}} (e^{x} + e^{x - 1})

What is expand (10^6K)/((is+10^2)^3) ?
The solution to expand (10^6K)/((is+10^2)^3) is