# How do you simplify i^21 + i^30?

Nov 13, 2015

$i - 1$

#### Explanation:

Observe that the powers of $i$ are cyclic:

${i}^{0} = 1$
${i}^{1} = i$
${i}^{2} = - 1$
${i}^{3} = - i$
${i}^{4} = 1$
$\ldots$

So, to find high powers of $i$, we can calculate the modulus $4$ Since $21 = 5 \cdot 4 + 1$, and $30 = 7 \cdot 4 + 2$, we have that ${i}^{21} = {i}^{1} = i$, and ${i}^{30} = {i}^{2} = - 1$. So,

${i}^{21} + {i}^{30} = i - 1$.