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

1.	Express the following complex numbers in the standard form a + ib : \(\frac{(1-i)^3}{1-i^3}\)
Answer» Given: ⇒ a + ib = \(\frac{(1-i)^3}{1-i^3}\) ⇒ a + ib = \(\frac{1^3-3(i)^2(i)+3(1)(i)^2-i^3}{1-i^2.i}\) We know that i²= -1 ⇒ a + ib = \(\frac{1-3i+3(-1)-i^2.i}{1-(-1)i}\) ⇒ a + ib = \(\frac{-2-3i-(-1)i}{1+i}\) ⇒ a + ib = \(\frac{-2-4i}{1+i}\) Multiplying and diving with 1-i ⇒ a + ib = \(\frac{-2-4i}{1+i} \times \ \frac{1-i}{1-i}\) ⇒ a + ib = \(\frac{-2(1-i)-4i(1-i)}{1^2-i^2}\) We know that i²= -1 ⇒ a+ib = \(\frac{-2+2i-4i+4i^2}{1-(-1)}\) ⇒ a + ib = \(\frac{-2-2i+4(-1)}{2}\) ⇒ a + ib = \(\frac{-6-2i}{2}\) ⇒ a + ib = -3-i ∴ The values of a, b are -3, -1.

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

Answer»

Given:

⇒ a + ib = \(\frac{(1-i)^3}{1-i^3}\)

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

We know that i²= -1

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

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

⇒ a + ib = \(\frac{-2-4i}{1+i}\)

Multiplying and diving with 1-i

⇒ a + ib = \(\frac{-2-4i}{1+i} \times \ \frac{1-i}{1-i}\)

⇒ a + ib = \(\frac{-2(1-i)-4i(1-i)}{1^2-i^2}\)

We know that i²= -1

⇒ a+ib = \(\frac{-2+2i-4i+4i^2}{1-(-1)}\)

⇒ a + ib = \(\frac{-2-2i+4(-1)}{2}\)

⇒ a + ib = \(\frac{-6-2i}{2}\)

⇒ a + ib = -3-i

∴ The values of a, b are -3, -1.