# How do you simplify a^-2times a^3timesa^4?

Mar 21, 2016

${a}^{-} 2 \times {a}^{3} \times {a}^{4} = {a}^{5}$

#### Explanation:

If $n$ is a positive integer then:

${a}^{n} = {\overbrace{a \times a \times . . \times a}}^{\text{n times}}$

So if $m$ and $n$ are both positive integers:

${a}^{m} \cdot {a}^{n} = {\overbrace{a \times a \times . . \times a}}^{\text{m times" xx overbrace(a xx a xx .. xx a)^"n times}}$

$= {\overbrace{a \times a \times . . \times a}}^{\text{m + n times}} = {a}^{m + n}$

If $a \ne 0$ then we can also define:

${a}^{- n} = \frac{1}{\underbrace{a \times a \times . . \times a}} _ \text{n times}$

In fact, if $a \ne 0$ then ${a}^{m} \cdot {a}^{n} = {a}^{m + n}$ regardless of whether $m$ and $n$ are positive, negative or zero.

$\textcolor{w h i t e}{}$
In our example,

${a}^{-} 2 \times {a}^{3} \times {a}^{4} = {a}^{- 2 + 3 + 4} = {a}^{5}$

Or if you prefer to see it a little slower:

${a}^{-} 2 \times {a}^{3} \times {a}^{4}$

$= \frac{1}{a \times a} \times \left(a \times a \times a\right) \times \left(a \times a \times a \times a\right)$

$= \frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{a \times a}}} \times a \times a \times a \times a \times a}{\textcolor{red}{\cancel{\textcolor{b l a c k}{\left(a \times a\right)}}}}$

$= a \times a \times a \times a \times a = {a}^{5}$