# How do you simplify 2^3xx2^-5?

${2}^{-} 2 = \frac{1}{4}$

#### Explanation:

We will use the rules of indices.

We know, ${a}^{m} \times {a}^{n} = {a}^{m + n}$ and ${a}^{-} m = \frac{1}{a} ^ m$
So, Here,

${2}^{3} \times {2}^{-} 5 = {2}^{3 + \left(- 5\right)} = {2}^{3 - 5} = {2}^{-} 2 = \frac{1}{4}$

We can use other laws of indices here.

${a}^{m} / {a}^{n} = {a}^{m - n}$

Using this,

${2}^{3} \times {2}^{-} 5 = {2}^{3} / {2}^{5} = {2}^{3 - 5} = {2}^{-} 2 = \frac{1}{4}$

And there are other ways too...

Hope this helps.

Mar 9, 2018

Using the same format as the question: $\textcolor{b l u e}{{2}^{- 2}}$

Two raise to the power of negative two.

#### Explanation:

$\textcolor{b l u e}{\text{The shortcut approach:}}$

Write as ${2}^{3 - 5} = {2}^{- 2}$

$\text{=====================================}$
$\textcolor{b l u e}{\text{Explaining why this works by using first principles}}$

$\textcolor{b r o w n}{\text{Before we start this is an important 'concept' (way of thinking)}}$

A whole number is what is called a 'rational number'. This means that it can be written in the fractional format of $\frac{a}{b}$ where $b$ is not 0. The way we do this is for example:

$1 , 2 , 3 , 4 , 5 \to \frac{1}{1} , \frac{2}{1} , \frac{3}{1} , \frac{5}{1}$ and so on. I will use this to emphasise a point.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b r o w n}{\text{Answering the question}}$

Note that another way of writing ${2}^{- 5}$ is $\frac{1}{2} ^ 5$. They both mean the same thing.

Putting this back together we have:

${2}^{3} \times \frac{1}{2} ^ 5$

$\frac{{2}^{3}}{1} \times \frac{1}{2} ^ 5$

$\frac{{2}^{3} \times 1}{1 \times {2}^{5}} = {2}^{3} / {2}^{5}$

$\frac{2 \times 2 \times 2}{2 \times 2 \times 2 \times 2 \times 2}$ we can 'split' this

$\underbrace{\frac{2 \times 2 \times 2}{2 \times 2 \times 2}} \textcolor{w h i t e}{\text{dd")xxcolor(white)("dd}} \frac{1}{2 \times 2}$
$\textcolor{w h i t e}{\text{ddd}} \downarrow$

color(white)("d.dd")ubrace(1color(white)("dddd.d")xxcolor(white)("dd")1/(2xx2))

$\textcolor{w h i t e}{\text{dddddddddd}} \frac{1}{2} ^ 2$

Another way of writing $\frac{1}{2} ^ 2$ is ${2}^{- 2}$

Also note that $\frac{1}{2} ^ 2 = \frac{1}{4} = {4}^{- 1}$

So ${4}^{- 1} = {2}^{- 2}$

Hope this helps!