Question

How do you simplify ${i^{36}}?$

Answer 1

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.