Question

How do you solve the expression \[{i^{42}}\]?

Answer 1

Hint:Iota (“i”) is known as the complex number whose value is minus root one, it was found by the mathematician to deal the negative sign under root, previously when this was not defined then if negative sign comes under root then there was no solution for that, but after this research complex terms can now easily be solved and tackled.

Formulae Used:
\[
\Rightarrow {i^{2n}} = -
1(where\,n\,represent\,every\,positive\,and\,negative\,\operatorname{int} eger\,) \\
\Rightarrow {i^{4n}} = 1(where\,n\,represent\,positive\,and\,negative\,\operatorname{int} eger)
\\
\]

Complete step by step solution:
The given question is \[{i^{42}}\]
Here we have to know the basic properties of iota which defines the value of iota on higher powers, the properties are:
\[
\Rightarrow {i^{2n}} = -
1(where\,n\,represent\,every\,positive\,and\,negative\,\operatorname{int} eger\,) \\
\Rightarrow {i^{4n}} = 1(where\,n\,represent\,positive\,and\,negative\,\operatorname{int} eger)
\\
\]
From the above properties we come to know that the power given in the question needs some evaluation and after using property we can direct answer the final value, on solving further we get:
\[ \Rightarrow {i^{42}} = {i^{40 + 2}} = {i^{40}} \times {i^2} = {i^{4 \times 10}} \times {i^2} = 1 \times (
- 1) = - 1\]
Here we have broken the power as per the properties and then obtained the final solution which is “-1”.

Additional Information: Dealing with the complex equation you have to be careful only when you're dealing in a higher degree equation because there the value of “iota” is given as for higher degree terms and accordingly the question needs to be solved.

Note: After the development of iota, research leads with the formulas associated and the properties like summation, subtraction, multiplication and division for the complex numbers. Graphs for complex numbers are also designed and the area under which the graph is drawn contains complex numbers only,but relation between complex and real numbers can be drawn.