Express the following complex numbers in the standard form a + ib :\(\

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

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

Answer»

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 .