# How do you simplify i^279?

Jan 14, 2016

An easy trick is to simply divide 279 by 4 and use the remainder to find your answer ...

#### Explanation:

Exponential powers of imaginary number $i$ cycle through only 4 possible results :

${i}^{1} = i$
${i}^{2} = - 1$
${i}^{3} = - i$
${i}^{4} = 1$

${i}^{5} = i$, etc...

Now, divide the exponent by 4 and find the remainder .

For example, ${i}^{6}$: $\frac{6}{4} = 1$ Remainder $2$
Next, simply look up the value of ${i}^{2}$ which is $- 1$
So, ${i}^{6} = - 1$

${i}^{279} : \frac{279}{4} = 69 \text{ with a Remainder} = 3$

${i}^{279} = {i}^{3} = - i$

hope that helped