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

1.	Express the following complex numbers in the standard form a + ib : \(\frac{(2+i)^3}{2+3i}\)
Answer» Given: ⇒ a+ ib = \(\frac{(2+i)^3}{2+3i}\) ⇒ a+ ib = \(\frac{2^3+i^3+3(2)^2(i)+3(i)^2(2)}{2+3i}\) ⇒ a+ ib = \(\frac{8+(i^2.i)+3(4)(i)+6i^2}{2+3i}\) We know that i²=-1 ⇒ a+ ib = \(\frac{8+(-1)i+12i+6(-1)}{2+3i}\) ⇒ a+ ib =\(\frac{2+11i}{2+3i}\) Multiplying and dividing with 2-3i ⇒ a+ ib = \(\frac{2+11i}{2+3i}\) x \(\frac{2-3i}{2-3i}\) ⇒ a+ ib = \(\frac{2(2-3i)+11i(2-3i)}{(2)^2-(3i)^2}\) ⇒ a+ ib = \(\frac{4-6i+22i-33i^2}{4-9i^2}\) We know that i²=-1 ⇒ a+ ib = \(\frac{4+16i-33(-1)}{4-9(-1)}\) ⇒ a+ ib = \(\frac{37+16i}{13}\) ∴ The values of a, b are \(\frac{37}{13}\) , \(\frac{16}{13}\).

Express the following complex numbers in the standard form a + ib : \(\frac{(2+i)^3}{2+3i}\)

Answer»

Given:

⇒ a+ ib = \(\frac{(2+i)^3}{2+3i}\)

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

⇒ a+ ib = \(\frac{8+(i^2.i)+3(4)(i)+6i^2}{2+3i}\)

We know that i²=-1

⇒ a+ ib = \(\frac{8+(-1)i+12i+6(-1)}{2+3i}\)

⇒ a+ ib =\(\frac{2+11i}{2+3i}\)

Multiplying and dividing with 2-3i

⇒ a+ ib = \(\frac{2+11i}{2+3i}\) x \(\frac{2-3i}{2-3i}\)

⇒ a+ ib = \(\frac{2(2-3i)+11i(2-3i)}{(2)^2-(3i)^2}\)

⇒ a+ ib = \(\frac{4-6i+22i-33i^2}{4-9i^2}\)

We know that i²=-1

⇒ a+ ib = \(\frac{4+16i-33(-1)}{4-9(-1)}\)

⇒ a+ ib = \(\frac{37+16i}{13}\)

∴ The values of a, b are \(\frac{37}{13}\) , \(\frac{16}{13}\).