Question

The value of the expression \[{\left( {1 + i } \right)^4} + {\left( {1 - i } \right)^4}\] is
A). \[8\]
B). \[8i \]
C). \[ - 8\]
D). \[32\]

Answer 1

Hint: In the given question, we have been given an expression involving the use of complex numbers. To solve it, we are going to simplify the expression to reduce it into the lowest form of the expression. Then we are going to apply the formula of the complex number simplification so as to get to the answer.

Formula used:
We are going to use the formula of whole square, which is,
\[{\left( {a \pm b} \right)^2} = {a^2} \pm 2ab + {b^2}\]

Complete step by step solution:
The given expression is \[{\left( {1 + i } \right)^4} + {\left( {1 - i } \right)^4}\].
It can be written as \[{\left( {{{\left( {1 + i } \right)}^2}} \right)^2} + {\left( {{{\left( {1 - i } \right)}^2}} \right)^2}\]
First, we will solve the whole square bracket,
\[{\left( {1 + i } \right)^2} = 1 + 2 \times 1 \times i + {\left( i \right)^2}\]
We know, \[i = \sqrt { - 1} \Rightarrow {i ^2} = - 1\]
So, \[{\left( {1 + i } \right)^2} = 1 + 2i + \left( { - 1} \right) = 2i \]
Similarly, for the second bracket,
\[{\left( {1 - i } \right)^2} = - 2i \]
Now, putting the values into the main function,
\[{\left( {2i } \right)^2} + {\left( { - 2i } \right)^2} = - 4 + \left( { - 4} \right) = - 8\]
Hence, the correct option is C.
complex number
Additional Information:
The “\[i \]” symbol multiplied with the constant is called the complex number. It has a value of \[\sqrt { - 1} \]. It is the imaginary part of the equation, as we know a negative number cannot be square rooted. There are a few properties of the number:
\[{i ^2} = - 1\]
\[{i ^3} = - i \]
\[{i ^4} = 1\]

Note: In the given question, we had to simplify a fraction to be written into the form of a standard complex number with their real and complex parts separated. Then we simplified the expression and solved for the answer. So, it is very important that we know the formulae, how to use them, when to use them, where to use them and the exact methodology for solving the question.