# How do you simplify i^247?

Jan 19, 2016

A great trick to solving this type of problem is understanding that there are only 4 possible solution ...

#### Explanation:

Raising the imaginary number $i$ to a power can only result in 4 possible solutions :

${i}^{0} = 1$
${i}^{1} = i$
${i}^{2} = - 1$
${i}^{3} = - i$

Here's the trick ... divide the exponent by the number 4 and calculate the remainder. Then, use the remainder as the new exponent .

In this example, $\frac{247}{4} = 61 \text{ with remainder } 3$

Answer: ${i}^{247} = {i}^{3} = - i$

hope that helped