# Indices examples

**Example 1**

What is `2^0` in a simpler form?

**Answer:**

We have two ways to remember this

## Method 1

Remember the picture "it doesnt matter how many people there are there is only one sun" which means

`2^0` is therefore `=1`

or

## Method 2

Try simple numbers you know first

`(10^2)/(10^2)=100/100=1`

and also

`(10^2)/(10^2)=10^2times10^-2=10^0` must`=1`

therefore `2^0=1`

**Example 2**

What is `49^(-1/2)` in a simpler form?

**Answer:**

Try simpler numbers we know first

We know

`10^-2=1/(10^2)`

so

`49^(-1/2)=1/(49^(1/2))`

and we also know the `1/2` denotes that it is root `2`.

So

`49^(1/2)` is `sqrt49`

and

`sqrt49=7`

Therfore

`1/49^(1/2)=1/sqrt49=1/7`

So the answer `=1/7`

**Example 3**

What is `64^(1/3)` in a simpler form

**Answer:**

## Method 1

`64^(1/3)` actually reads

Work out what we multiply by itself `3` times to get `64`

Which would be `xtimesxtimesx=64`

this is `4times4times4=64`

Therefore `64^(1/3)=4`

## Method 2

Try simpler numbers you know

You might recognise that

`9^(1/2)=sqrt9=3`

and

`27^(1/3)=root(3)(27)=3`

Therefore

`64^(1/3)=root(3)(64)=4`

**Example 4**

What is `1^9`

**Answer:**

Try simpler numbers you know

We know

`2^2=2times2=4`

Therefore

`1^9=1times1times1times1times1times1times1times1times1`

`=1`

Therefore

`1^9=1`

**Example 5**

What is `(1\2/5)^3`

**Answer:**

The first thing to do is turn

`1\2/5` into a fraction

See mammoth memory fractions

`1\2/5=7/5`

So `(1\2/5)^3` becomes `(7/5)^3`

"Try simple numbers you know first"

Try `(4/2)^3=2^3=2times2times2=8`

is this the same as

`4^3/2^3=(4times4times4)/(2times2times2)`

`=(cancel4\ ^2timescancel4\ ^2timescancel4\ ^2)/(cancel2timescancel2timescancel2)`

`=2times2times2=8`

Yes it is

Therefore

`(7/5)^3=7^3/5^3`

`=(7times7times7)/(5times5times5)`

`=343/275`

`1.247` to `3` decimal places.

**Example 6**

What is `16^(3/2)`

To tackle this we "always split the power into a root and a power"

`16^(3/2)=` either `(16^3)^(1/2)` or `(16^(1/2))^3`

Lets go for

`(16^(1/2))^3`

Which is `(root(2)(16))^3`

Which is `4^3`

and this is `4times4times4`

`=16times4`

`=64`

**Example 7**

Simplify `(4x^-2)/y^0`

We must remember the picture

"It doesn't matter how many people there are there is only one sun" which means

`y^0=1`

Therefore

`(4x^-2)/y^0`

`= (4x^-2)/1`

Which is just `4x^-2`

But "trying a simpler number we know first"

`10^-2=1/(10^2)`

Therefore

`4x^-2`

is the same as

`4/x^2`

**Example 8**

Simplify `18x^4y^5div3xy^4`

This would be the same as

`(18x^4y^5)/(3xy^4)`

and can be re-written as

`(18timesxtimesxtimesxtimesxtimesytimesytimesytimesytimesy)/(3xtimesytimesytimesytimesy)`

and if we cancel out we get

`(18timesxtimesxtimesxtimescancelxtimesytimescancelytimescancelytimescancelytimescancely)/(3timescancelxtimescancelytimescancelytimescancelytimescancely)`

`=(18timesxtimesxtimesxtimesy)/3`

`=(18x^3y)/3`

Or

`6x^3y`