Express the following complex numbers in the standard form a + ib : 5+√2i/1-√2i

Question

Express the following complex numbers in the standard form a + ib :\(\frac{5+\sqrt2i}{1-\sqrt2i}\)

Answer 1

Given: 

⇒ a + ib = \(\frac{5+\sqrt2i}{1-\sqrt2i}\)

Multiplying and dividing with 1+√2i

⇒ a + ib =  \(\frac{5+\sqrt2i}{1-\sqrt2i} \times \frac{1+\sqrt2i}{1+\sqrt2i}\) 

⇒ a + ib =  \(\frac{5(1+\sqrt2i)+\sqrt2i(1+\sqrt2i)}{1^2-(\sqrt2i)^2}\) 

⇒ a + ib =  \(\frac{5+5\sqrt2i+\sqrt2i+2i^2}{1-2i^2}\)

We know that i²=-1

⇒ a + ib = \(\frac{5+6\sqrt2i+2(-1)}{1-2(-1)}\) 

⇒ a + ib =  \(\frac{3+6\sqrt2i}{3}\) 

⇒ a + ib = 1 + 2√2i

∴ The values of a, b are 1, 2√2 .