Question

if √a+ib = x-iy. Prove that √a+ib = x+iy​

Answer 1

Answer:

Starting with the original equation

√a+ib=x+iy

equating the real terms of this equation note that

√a=x, we'll use this relationship near the end of the proof

Now we'll go back to the original equation and square both sides

(√a+ib)2=(x+iy)2

(√a+ib)(√a+ib)=(x+iy)(x+iy)

a+2√abi+i2b2=x2+2xyi+i2y2

noting i2=-1 implies

a+2√abi-b2=x2+2xyi-y2

equating the imaginary term on either side of the equation we have

2√ab=2xy

b=xy/√a, we'll use this relationship in just a second

Back to the original equation, again

√a+ib=x+iy

subtracting -2ib from either side

√a-ib=x+iy-2ib=x+i(y-2b)

Now lets substitute in the result we obtained by squaring the original equation, namely

b=xy/√a

yields

√a-ib=x+i(y-2xy/√a)=x+iy(1-2x/√a)

finally, lets substitute the intial result we got at the beginning of this proof ie √a=x

√a-ib=x+iy(1-2x/x)=x+iy(1-2)

Therefor

√a-ib=x-iy

Good Luck.

Cheers

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