Represent in the form of a …

Ans. \(let \space (a+ib) = i^{3\over 2}\)

Squaring Both Sides, We get 

\(=> a^2 +b^2.i^2 +2abi = i^3\)

\(=> a^2 -b^2 +2abi = -i \space \space \space \space \space \space \space \space \space \space [i^2 = -1]\)

On Comparing Both sides, We get 

\(a^2-b^2 = 0 \space \space \space \space \space \space \space and \space \space \space \space \space \space 2ab =-1\)

=> \(a^2 = b^2 \space \space \space ...(1) \space \space \space \space \space and \space \space a = {-1\over 2b} \space \space \space \space \space ... (2)\)

Put value of a in (1)

=> \(({-1\over 2b})^2 = b^2 => {1\over 4b^2} = b^2 => {1\over 4} = b^4 \)

=> \(b^2 = {1\over 2} => b = {1\over \sqrt 2}\)

Put value of b in (2), we get \(a = {-1\over 2 \times {1\over \sqrt 2}} => a = {-1\over \sqrt2}\)

So
=> \({-1\over \sqrt2}+{1\over \sqrt 2}i = i^{3\over 2}\)