Question

express in a+ib form of (3-i) ^4​

Answer 1

Explanation :

$\tt \red\star \: \: (3 - i) {}^{4}$

Applying binomial theorem,

$\boxed{ \tt \: (a + b) {}^{n} = \tt\sum\limits_{i = 0}^{n} \bigg( \cfrac{n}{i} \bigg)a {}^{(n - i)} b {}^{i}} \: \: \green \star$

Where,

• a = 3

• b = -i

Substitute all values in above formula,

$\to \tt \: \sum\limits_{i = 0}^{4} \bigg( \cfrac{4}{i} \bigg) \times 3 {}^{(4 - i)} ( - i) {}^{i} \\ \\ \\ \to \tt \: \cfrac{4!}{0!(4 - 0)!} \times 3 {}^{4 } ( - i) {}^{0} + \cfrac{4!}{1!(4 - 1)!} \times 3 {}^{3} ( - i) {}^{1} + \cfrac{4!}{2!(4 - 2)! } 3 {}^{2} ( - i) {}^{3} + \cfrac{4!}{3!(4 - 3)! } 3 {}^{1} ( - i) {}^{3} + \cfrac{4!}{4!(4 - 4)! } 3 {}^{0} ( - i) {}^{4} \\ \\ \\ \to \tt \:81 - 108i - 54 + 12i + 1 \\ \\ \\ \to \boxed{\tt \: 28 - 96i \: }\blue \star$