# How do you simplify i^656?

Jan 13, 2016

${i}^{656} = 1$

#### Explanation:

We will use the following:

• ${i}^{2} = - 1 \implies {\left({i}^{2}\right)}^{2} = {\left(- 1\right)}^{2} = 1$

• ${\left({x}^{a}\right)}^{b} = {x}^{a b}$

As $656 = 4 \cdot 164$ this means

${i}^{656} = {i}^{4 \cdot 164} = {\left({i}^{4}\right)}^{164} = {1}^{164} = 1$

In general, it is easy to evaluate ${i}^{n}$ by "pulling out" the greatest multiple of $4$ possible from the exponent. Either the exponent will be a multiple of $4$, as above, or we will be left with $i$, ${i}^{2}$, or ${i}^{3}$. For example:

${i}^{23} = {i}^{20} {i}^{3} = {i}^{4 \cdot 5} {i}^{3} = {\left({i}^{4}\right)}^{5} {i}^{3} = {i}^{3} = {i}^{2} i = \left(- 1\right) i = - i$