How do you simplify \[{i^{36}}?\]

Hint:This question involves the arithmetic operation of addition/ subtraction/ multiplication/ division. Also, we need to know the algebraic formula with the involvement of exponent components. We need to know the value \[{i^2}\] to solve the given problem or we can use a scientific calculator in complex mode to find the value \[{i^2}\]. We need to know the multiplication process with two different sign terms.

Complete step by step solution:
The given question is shown below,
\[{i^{36}} = ? \to \left( 1 \right)\]
The above equation can also be written as,
\[
\left( 1 \right) \to {i^{36}} = ? \\
{i^{36}} = {i^{4 \times 9}} \to \left( 2 \right) \\
\]
We know that,
\[{i^{a \times b}} = {\left( {{i^a}} \right)^b} \to \left( 3 \right)\]
By using the equation\[\left( 3 \right)\], the equation\[\left( 2 \right)\]can also be written as,
\[
\left( 2 \right) \to {i^{36}} = {i^{4 \times 9}} \\
{i^{4 \times 9}} = {\left( {{i^4}} \right)^9} \to \left( 4 \right) \\
\]
We know that,
\[{i^2} = - 1\]
Take square on both sides of the above equation, we get
\[
{\left( {{i^2}} \right)^2} = {\left( { - 1} \right)^2} \\
{i^4} = 1 \to \left( 5 \right) \\
\]
Let’s substitute the equation\[\left( 5 \right)\]in the equation\[\left( 4 \right)\], we get
\[
\left( 4 \right) \to {i^{4 \times 9}} = {\left( {{i^4}} \right)^9} \\
{i^{4 \times 9}} = {\left( 1 \right)^9} \\
\]
We know that,
\[1 \times 1 \times 1....... = 1\]
So, we get
\[{i^{4 \times 9}} = 1\]
So, the final answer is,
\[{i^{36}} = 1\]

Note: The above equation can also be solved by using a scientific calculator in complex mode. This question involves the arithmetic operation of addition/ subtraction/ multiplication/ division. Note that \[{i^2}\] and \[{\left( { - i} \right)^2}\] is equal to\[ - 1\]. Remember the algebraic formula with the involvement of exponent components. Also, note that\[{1^n}\] is equal to the value of \[1\].
Remember the following things when multiplying different sign terms,
1) When a positive term is multiplied with a positive term the final answer would be a positive term.
2) When a negative term is multiplied with a negative number the final answer would be a
positive term.
3) When a positive term is multiplied with a negative term the final answer would be a
negative term.