What is 1/((3-8i*x)^5) ?

The solution to 1/((3-8i*x)^5) is

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simplify 1/((3-8i*x)^5)

simplify

\frac{1}{( 3 - 8 i \cdot x ) ^{5}}

\frac{243 - 17280 x ^{2} + 61440 x ^{4}}{1073741824 x ^{10} + 754974720 x ^{8} + 212336640 x ^{6} + 29859840 x ^{4} + 2099520 x ^{2} + 59049} + \frac{3240 x - 46080 x ^{3} + 32768 x ^{5}}{1073741824 x ^{10} + 754974720 x ^{8} + 212336640 x ^{6} + 29859840 x ^{4} + 2099520 x ^{2} + 59049} i

Solution steps

\frac{1}{( 3 - 8 i x ) ^{5}}

Expand

(3 - 8 i x)^{5} : 243 - 3240 i x - 17280 x^{2} + 46080 i x^{3} + 61440 x^{4} - 32768 i x^{5}

= \frac{1}{243 - 3240 i x - 17280 x ^{2} + 46080 i x ^{3} + 61440 x ^{4} - 32768 i x ^{5}}

Rewrite

243 - 3240 i x - 17280 x^{2} + 46080 i x^{3} + 61440 x^{4} - 32768 i x^{5}

in standard complex form:

(243 - 17280 x^{2} + 61440 x^{4}) + (- 3240 x + 46080 x^{3} - 32768 x^{5}) i

= \frac{1}{( 243 - 17280 x ^{2} + 61440 x ^{4} ) + ( - 3240 x + 46080 x ^{3} - 32768 x ^{5} ) i}

Multiply by the conjugate

\frac{( 243 - 17280 x ^{2} + 61440 x ^{4} ) + ( 3240 x - 46080 x ^{3} + 32768 x ^{5} ) i}{( 243 - 17280 x ^{2} + 61440 x ^{4} ) + ( 3240 x - 46080 x ^{3} + 32768 x ^{5} ) i}

= \frac{1 \cdot ( ( 243 - 17280 x ^{2} + 61440 x ^{4} ) + ( 3240 x - 46080 x ^{3} + 32768 x ^{5} ) i )}{( ( 243 - 17280 x ^{2} + 61440 x ^{4} ) + ( - 3240 x + 46080 x ^{3} - 32768 x ^{5} ) i ) ( ( 243 - 17280 x ^{2} + 61440 x ^{4} ) + ( 3240 x - 46080 x ^{3} + 32768 x ^{5} ) i )}

Simplify

\frac{1 \cdot ( ( 243 - 17280 x ^{2} + 61440 x ^{4} ) + ( 3240 x - 46080 x ^{3} + 32768 x ^{5} ) i )}{( ( 243 - 17280 x ^{2} + 61440 x ^{4} ) + ( - 3240 x + 46080 x ^{3} - 32768 x ^{5} ) i ) ( ( 243 - 17280 x ^{2} + 61440 x ^{4} ) + ( 3240 x - 46080 x ^{3} + 32768 x ^{5} ) i )} : \frac{243 - 17280 x ^{2} + 61440 x ^{4} + i ( 3240 x - 46080 x ^{3} + 32768 x ^{5} )}{1073741824 x ^{10} + 754974720 x ^{8} + 212336640 x ^{6} + 29859840 x ^{4} + 2099520 x ^{2} + 59049}

= \frac{243 - 17280 x ^{2} + 61440 x ^{4} + i ( 3240 x - 46080 x ^{3} + 32768 x ^{5} )}{1073741824 x ^{10} + 754974720 x ^{8} + 212336640 x ^{6} + 29859840 x ^{4} + 2099520 x ^{2} + 59049}

Rewrite in standard complex form:

\frac{243 - 17280 x ^{2} + 61440 x ^{4}}{1073741824 x ^{10} + 754974720 x ^{8} + 212336640 x ^{6} + 29859840 x ^{4} + 2099520 x ^{2} + 59049} + \frac{3240 x - 46080 x ^{3} + 32768 x ^{5}}{1073741824 x ^{10} + 754974720 x ^{8} + 212336640 x ^{6} + 29859840 x ^{4} + 2099520 x ^{2} + 59049} i

= \frac{243 - 17280 x ^{2} + 61440 x ^{4}}{1073741824 x ^{10} + 754974720 x ^{8} + 212336640 x ^{6} + 29859840 x ^{4} + 2099520 x ^{2} + 59049} + \frac{3240 x - 46080 x ^{3} + 32768 x ^{5}}{1073741824 x ^{10} + 754974720 x ^{8} + 212336640 x ^{6} + 29859840 x ^{4} + 2099520 x ^{2} + 59049} i

v (2, - 1) y f (4, - 1)

\frac{s ^{4} + s + 1}{s ^{4} ( s - 1 )}

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

\frac{k ( 3 k - 1 )}{2} + (3 k - 1)

\frac{25 ( s + 0.03 )}{( s + 0.4 ) ( s ^{2} - 0.36 s + 0.16 )}