Question

What is the value of \[\left( 1+i \right)\left( 1+{{i}^{2}} \right)\left( 1+{{i}^{3}} \right)\left( 1+{{i}^{4}} \right)\]

Answer 1

Hint: To solve this type of problem we have to know the direct value of \[{{i}^{2}}\]. After that, convert the Iota power terms in the form of \[{{i}^{2}}\] and solve the expression. After simplification solve all the terms to get the final answer.

Complete step-by-step solution -
Given \[\left( 1+i \right)\left( 1+{{i}^{2}} \right)\left( 1+{{i}^{3}} \right)\left( 1+{{i}^{4}} \right)\]
We know that \[{{i}^{2}}=-1\] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)
Now converting the i power terms in the form of \[{{i}^{2}}\]
\[{{i}^{3}}=i.{{i}^{2}}=-i\] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)
\[{{i}^{4}}={{i}^{2}}.{{i}^{2}}=-1.-1=1\] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)
Now substituting (1) (2) (3) in the above expression we get,
\[\left( 1+i \right)\left( 1-1 \right)\left( 1-i \right)\left( 1+1 \right)\]
Now further simplifying the terms we get,
\[\left( 1-{{i}^{2}} \right)\left( 0 \right)\left( 2 \right)\]
The value for the above expression is 0.

Note: We can solve this type of problem if we know the value of \[{{i}^{2}}\]. The value of the above expression is zero because the term \[\left( 1+{{i}^{2}} \right)\] is zero so if we multiply any term with zero we get the answer as zero. Be careful while doing calculations. Here we can use an alternate method by multiplying all the brackets and using the power of iota to simplify the expression.