Question

Solve ${{\left( 1+i \right)}^{4}}+{{\left( 1-i \right)}^{4}}=$

Answer 1

Hint: We will use the general algebra of simplification to solve this question; also we use the values of iota with different powers to find the value of given expression. The following algebraic formulas will be used to simplify the given expression-
$\begin{align}
  & {{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab \\
 & {{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab \\
\end{align}$

Complete step by step answer:
We have been given an expression ${{\left( 1+i \right)}^{4}}+{{\left( 1-i \right)}^{4}}$.
We have to find the value of the given expression.
To find the value of the given expression first let us simplify the expression we have
$\Rightarrow {{\left[ {{\left( 1+i \right)}^{2}} \right]}^{2}}+{{\left[ {{\left( 1-i \right)}^{2}} \right]}^{2}}$
Now we know that the algebraic identities of simplification are
$\begin{align}
  & {{\left( a+b \right)}^{2}}={{a}^{2}}+{{b}^{2}}+2ab \\
 & {{\left( a-b \right)}^{2}}={{a}^{2}}+{{b}^{2}}-2ab \\
\end{align}$
So, when we apply the formula on the given expression we get
$\Rightarrow {{\left[ 1+{{i}^{2}}+2i \right]}^{2}}+{{\left[ 1+{{i}^{2}}-2i \right]}^{2}}$
Now, we know that the value of imaginary number iota with power will be ${{i}^{2}}=-1$
So, by putting the value of ${{i}^{2}}$ in the above equation we get
$\begin{align}
  & \Rightarrow {{\left[ 1+\left( -1 \right)+2i \right]}^{2}}+{{\left[ 1+\left( -1 \right)-2i \right]}^{2}} \\
 & \Rightarrow {{\left[ 1-1+2i \right]}^{2}}+{{\left[ 1-1-2i \right]}^{2}} \\
\end{align}$
Now, solving further we get
$\begin{align}
  & \Rightarrow {{\left[ 2i \right]}^{2}}+{{\left[ -2i \right]}^{2}} \\
 & \Rightarrow 4{{i}^{2}}+4{{i}^{2}} \\
\end{align}$
Now, again putting the value ${{i}^{2}}=-1$ in the above equation we get
$\begin{align}
  & \Rightarrow 4\left( -1 \right)+4\left( -1 \right) \\
 & \Rightarrow -4-4 \\
 & \Rightarrow -8 \\
\end{align}$

So, we get ${{\left( 1+i \right)}^{4}}+{{\left( 1-i \right)}^{4}}=-8$

Note: To solve these types of questions we need to know the basics of iota $i$ that it is used to represent the imaginary part of a complex number of the form $a+ib$, where $a\And b$ are real numbers and $i$ is the imaginary number. Also the values of different powers of $i$ are different so one must have knowledge about the values. The alternative way to solve this question is by applying the direct formula of ${{a}^{4}}+{{b}^{4}}$. The simplified form of ${{a}^{4}}+{{b}^{4}}$ is as follows:
${{a}^{4}}+{{b}^{4}}={{\left( {{a}^{2}}+{{b}^{2}} \right)}^{2}}-2{{a}^{2}}{{b}^{2}}$.