Express the following complex numbers in the standard form a + ib :(1+i)(1+√3i)/1-i

Question

Express the following complex numbers in the standard form a + ib :  \(\frac{(1+i)(1+\sqrt{3}i)}{1-i}\) 

Answer 1

Given: 

⇒ a + ib = \(\frac{(1+i)(1+\sqrt{3}i)}{1-i}\)  

⇒ a + ib = \(\frac{1(1+\sqrt{3}i)+i(1+\sqrt{3}i)}{1-i}\)

⇒ a + ib = \(\frac{1+\sqrt{3}i+i+\sqrt{3}i^2}{1-i}\)

We know that i²=-1 

⇒ a + ib = \(\frac{1+(\sqrt{3}+1)i+\sqrt{3}(-1)}{1-i}\)

⇒ a + ib = \(\frac{(1-\sqrt{3})+(1+\sqrt{3})i}{1-i}\)

Multiplying and dividing with 1+i

∴ The values of a, b are -√3,1.