Question

After simplification the complex number (3 + i)(1+ 3i)² reduces to the form...
a) 3 +3i b) -9 c)-30 + 10i d) 10 + 30i
Ans, options:
1. 3+3i
2. -9
3. - 30 + 1Oi
4. 10+ 30i​

Answer 1

Given :

• The given complex numbers is (3+i)(1+3i)²

To Find :

• The value of (3+i)(1+3i)²=?

Formula :

The values of complex numbers (iota) :

$\bullet \sf \: {i}^{0} = 1$

$\bullet \sf \: {i}^{1} = i$

$\bullet \sf \: {i}^{2} = - 1$

$\bullet \sf \: {i}^{3} = {i}^{2} \times i = - i$

$\bullet \sf \: {i}^{4} = {( {i}^{2} )}^{2} = 1$

Solution :

The given expression of the complex number is (3+i)(1+3i)²

$\implies \sf \: (3 + i) {(1 + 3i)}^{2}$

By using formula :

$\bullet \sf \: {(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab$

$\\ \implies \sf \: (3 + i) \bigg( {(1)}^{2} + {(3i)}^{2} + 2 \times 1 \times 3i \bigg)$

$\implies \sf \: (3 + i)(1 - 9 + 6i)$

$\implies \sf \: (3 + i)( - 8 + 6i)$

$\implies \sf \: 3( - 8 + 6i) + i( - 8 + 6i)$

$\implies \sf \: - 24 + 18i - 8i + 6 {i}^{2}$

$\implies \sf \: - 24 + 10i - 6$

$\implies \sf \: - 30 + 10i$

$\implies \boxed{ \bf{ - 30 + 10i}}$

The value of this equation is -30+10i.