What is 1/((1+isqrt(3))^4) ?

The solution to 1/((1+isqrt(3))^4) is -1/32+(sqrt(3))/(32)i

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simplify 1/((1+isqrt(3))^4)

simplify

\frac{1}{( 1 + i 3 ) ^{4}}

- \frac{1}{32} + \frac{3}{32} i

Solution steps

\frac{1}{( 1 + i 3 ) ^{4}}

Expand

(1 + i 3)^{4} : - 83 i - 8

= \frac{1}{- 8 3 i - 8}

Rewrite

- 83 i - 8

in standard complex form:

- 8 - 83 i

= \frac{1}{- 8 - 8 3 i}

Multiply by the conjugate

\frac{- 8 + 8 3 i}{- 8 + 8 3 i}

= \frac{1 \cdot ( - 8 + 8 3 i )}{( - 8 - 8 3 i ) ( - 8 + 8 3 i )}

Simplify

\frac{1 \cdot ( - 8 + 8 3 i )}{( - 8 - 8 3 i ) ( - 8 + 8 3 i )} : \frac{- 8 + 8 3 i}{256}

= \frac{- 8 + 8 3 i}{256}

Apply the fraction rule:

\frac{a + bi}{c} = \frac{a}{c} + \frac{b}{c} i

= \frac{- 8}{256} + \frac{8 3}{256} i

Simplify

\frac{- 8}{256} + \frac{8 3}{256} i : - \frac{1}{32} + \frac{3}{32} i

= - \frac{1}{32} + \frac{3}{32} i

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What is 1/((1+isqrt(3))^4) ?
The solution to 1/((1+isqrt(3))^4) is -1/32+(sqrt(3))/(32)i