Question

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

Answer 1

`2^-2 = 1/4`

Explanation:

We will use the rules of indices.

We know, `a^m xx a^n = a^(m + n)` and `a^-m = 1/a^m`
So, Here,

`2^3 xx 2^-5 = 2^(3 + (-5)) = 2^(3 - 5) = 2^-2 = 1/4`

We can use other laws of indices here.

`a^m/a^n = a^(m-n)`

Using this,

`2^3 xx 2^-5 = 2^3/2^5 = 2^(3 - 5) = 2^-2 = 1/4`

And there are other ways too...

Hope this helps.

Answer 2

Using the same format as the question: `color(blue)(2^(-2))`

Two raise to the power of negative two.

Explanation:

`color(blue)("The shortcut approach:")`

Write as `2^(3-5) = 2^(-2)`

`"====================================="`
`color(blue)("Explaining why this works by using first principles")`

`color(brown)("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 `a/b` where `b` is not 0. The way we do this is for example:

`1,2,3,4,5 ->1/1,2/1,3/1,5/1` and so on. I will use this to emphasise a point.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(brown)("Answering the question")`

Note that another way of writing `2^(-5)` is `1/2^5`. They both mean the same thing.

Putting this back together we have:

`2^3xx1/2^5`

`(2^3)/1xx1/2^5`

`(2^3xx1)/(1xx2^5) = 2^3/2^5`

`(2xx2xx2)/(2xx2xx2xx2xx2)` we can 'split' this

`ubrace((2xx2xx2)/(2xx2xx2))color(white)("dd")xxcolor(white)("dd")1/(2xx2)`
`color(white)("ddd")darr`

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

`color(white)("dddddddddd")1/2^2`

Another way of writing `1/2^2` is `2^(-2)`

Also note that `1/2^2=1/4=4^(-1)`

So `4^(-1)=2^(-2)`

Hope this helps!