# How do you simplify i^10+i^2?

Dec 3, 2016

The expression simplifies to $- 2$.

#### Explanation:

${i}^{10} + {i}^{2}$

Use the exponent rule ${\left({x}^{a}\right)}^{b} = {x}^{a b}$ and change the ${i}^{10}$ term to an ${i}^{2}$ term by dividing the exponent $\textcolor{b l u e}{10}$ by $\textcolor{red}{2} = \textcolor{m a \ge n t a}{5}$.

${i}^{\textcolor{b l u e}{10}} = {\left({i}^{\textcolor{red}{2}}\right)}^{\textcolor{m a \ge n t a}{5}}$

${\left({i}^{2}\right)}^{5} + {i}^{2}$

Recall that ${i}^{2} = - 1$

${\left(- 1\right)}^{5} + \left(- 1\right)$

$- 1$ raised to an odd power is equal to $- 1$ (and btw, $- 1$ raised to an even power is equal to $1$)

$\left(- 1\right) + \left(- 1\right) = - 2$