Question

Find the value of \[{\left( {1 + i} \right)^5}{\left( {1 - i} \right)^5}\].
A. -8
B. \[8i\]
C. 8
D. 32

Answer 1

Hint: First we will apply the indices formula and algebraic identity to simplify the given expression. Then we will put the value of \[{i^2}\]. After simplify we will get the value of \[{\left( {1 + i} \right)^5}{\left( {1 - i} \right)^5}\].

Formula used:
1. \[{a^m} \cdot {b^m} = {\left( {ab} \right)^m}\]
2. \[\left( {a + b} \right)\left( {a - b} \right) = {a^2} - {b^2}\]
3. \[{i^2} = - 1\]

Complete step by step solution:
Given expression is
\[{\left( {1 + i} \right)^5}{\left( {1 - i} \right)^5}\]
Applying the formula of indices
\[ = {\left[ {\left( {1 + i} \right)\left( {1 - i} \right)} \right]^5}\] [ Since \[{a^m} \cdot {b^m} = {\left( {ab} \right)^m}\]]
Applying the identity formula
\[ = {\left[ {{1^2} - {i^2}} \right]^5}\]
Substitute the value of \[{i^2}\]
\[ = {\left[ {1 - \left( { - 1} \right)} \right]^5}\] [Since \[{i^2} = - 1\]]
\[ = {\left[ {1 + 1} \right]^5}\]
\[ = {2^5}\]
\[ = 32\]
Hence option D is the correct option.

Addition information: i is a complex number. The value of i is \[\sqrt{-1}\]. If the power of i is a multiple of 4 then the value of the complex number will be 1.

Note: Many students solve the question by using binomial expansion which is a lengthy process. In this process, we will apply the binomial expansion formula on \[{\left( {1 + i} \right)^5}\] and \[{\left( {1 - i} \right)^5}\]. After that use the combination formula \[{^n}{C_r}=\dfrac{n!}{(n-r)!r!}\] to simplify them and also substitute \[{i^2} = - 1,{i^3} = - i,{i^4} = 1,{i^5} = i\]. Then multiply the terms to get the desired answer.