Express the following complex numbers in the standard form a + ib :(2+i)^3/2+3i

Question

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

Answer 1

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 \(\frac{37}{13}\) , \(\frac{16}{13}\).