# How do you simplify i^42?

Nov 9, 2015

The imaginary number $i$ can only take on 4 values when raised by a positive integer exponent.

#### Explanation:

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

Then, it simply cycles through the same values all over.

To find ${i}^{42}$ just divide the exponent by 4 and find the remainder :

$\frac{42}{4} = 10$ with a remainder of 2. So, the value 2 is the exponent.

Answer: ${i}^{2} = - 1$

Hope that helped