The solution to (i-a)^{-1} is -a/(a^2+1)-1/(a^2+1)i

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simplify (i-a)^{-1}

Solution

simplify

(i - a)^{- 1}

- \frac{a}{a ^{2} + 1} - \frac{1}{a ^{2} + 1} i

Solution steps

(i - a)^{- 1}

Apply exponent rule:

a^{- 1} = \frac{1}{a}

(i - a)^{- 1} = \frac{1}{i - a}

= \frac{1}{i - a}

Rewrite

i - a

in standard complex form:

- a + i

= \frac{1}{- a + i}

Multiply by the conjugate

\frac{- a - i}{- a - i}

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

Simplify

\frac{1 \cdot ( - a - i )}{( - a + i ) ( - a - i )} : \frac{- a - i}{a ^{2} + 1}

= \frac{- a - i}{a ^{2} + 1}

Apply the fraction rule:

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

= \frac{- a}{a ^{2} + 1} + \frac{- 1}{a ^{2} + 1} i

Simplify

\frac{- a}{a ^{2} + 1} + \frac{- 1}{a ^{2} + 1} i : - \frac{a}{a ^{2} + 1} - \frac{1}{a ^{2} + 1} i

= - \frac{a}{a ^{2} + 1} - \frac{1}{a ^{2} + 1} i

2 e^{z + 2}

\frac{6 n ^{- 3} y}{2 n ^{- 1} y ^{- 3}}

\frac{1}{( b - c ) ^{2}} + \frac{1}{( c - a ) ^{2}} + \frac{1}{( a - b ) ^{2}}

(y^{- 5})^{3}

(3 m + \frac{1}{5})^{2}