# How do you simplify 7i^40 - 9i^100?

Nov 10, 2015

$7 {i}^{40} - 9 {i}^{100} = - 2$

#### Explanation:

We will use two facts.

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

2) ${i}^{2} = - 1 \implies {i}^{4} = {\left({i}^{2}\right)}^{2} = 1$

Applying (1), we get

$7 {i}^{40} - 9 {i}^{100} = 7 {i}^{4 \cdot 10} - 9 {i}^{4 \cdot 25} = 7 {\left({i}^{4}\right)}^{10} - 9 {\left({i}^{4}\right)}^{25}$

Then, by (2),

$7 {\left({i}^{4}\right)}^{10} - 9 {\left({i}^{4}\right)}^{25} = 7 \cdot {1}^{10} - 9 \cdot {1}^{25} = 7 - 9 = - 2$

So our final result is

$7 {i}^{40} - 9 {i}^{100} = - 2$