# Standard form difficult examples

**Example 1**

Work out the following giving your answer in standard form

`(3.2times10^-5)times(4.8times10^7)`

## Method 1

Is to actually carry out the multiplication

`(3.2times10^-5)times(4.8times10^7)`

is the same as

`3.2/(cancel10timescancel10timescancel10timescancel10timescancel10)`

`times4.8times10times10timescancel10timescancel10timescancel10timescancel10timescancel10`

`=3.2times4.8times10times10`

`=1,536`

Now place in standard form

remember move decimal place to the left and `+1` to the power

`=1.536times10^3`

## Method 2

An alternative way to calculate this is to use indices. What is a similar calculation? What is:

`10^2times10^-3=(10times10)/(10times10times10)=1/10=10^-1`

Therefore

`10^2times10^-3=10^(2-3)=10^1`

So

`(3.2times10^-5)times(4.8times10^7)`

`=3.2times4.8times10^-5times10^7`

`=3.2times4.8times10^(-5+7)`

`=3.2times4.8times10^2`

`=3.2times4.8times10times10`

`=1,536`

and in standard form that equals

remember move decimal place to left and `+1` to the power

`1.536times10^3`

**Example 2**

Work out the following giving your answer in standard format:

`sqrt((6.8times10^9-3.8times10^8)/(6.8times10^9times3.8times10^8))`

## Method 1

Is to actually carry out each calculation but achieve this step by step

`6.8times10^9-3.8times10^8`

is the same as

`6.8times10times10times10times10times10times10times10times10times10`

`-3.8times10times10times10times10times10times10times10times10`

`=6,800,000,000-380,000,000`

which is

`1` | |||||||||||

`=` | `6,` | `8` | `0` | `0,` | `0` | `0` | `0,` | `0` | `0` | `0` | |

`4` | |||||||||||

`-` | `cancel3` | `8` | `0,` | `0` | `0` | `0,` | `0` | `0` | `0` | ||

`6,` | `4` | `2` | `0,` | `0` | `0` | `0,` | `0` | `0` | `0` |

and

`6.8times10^9times3.8times10^8`

is the same as

`6.8times10times10times10times10times10times10times10times10times10`

`times3.8times10times10times10times10times10times10times10times10`

and the same as

`6.8times3.8times10^17`

which is

`25.84times10^17`

so the calculation becomes

`sqrt((6,420,cancel(000),cancel(000))/(2584,000,000,000,cancel(000),cancel(000)))`

or

`sqrt((6,420)/(2584,000,000,000))`

`=sqrt0.00000000248452`

`=0.00004984`

or in standard form

remember move decimal to right and `-1` to the power

`4.984times10^-5`

## Method 2

An alternative way is to add or take the indices.

Start with the top line

`6.8times10^9-3.8times10^8`

We have to make the powers the same when we subtract.

Remember moving decimals to the right and `-1` to the power.

so

`6.8times10^9=68times10^8`

this becomes

`68times10^8-3.8times10^8`

which is

`(68-3.8)times10^8`

`=64.2times10^8`

Now for the bottom line

`6.8times10^9times3.8times10^8`

which is

`6.8times3.8times10^9times10^8`

this becomes

`6.8times3.8times10^17`

`25.84times10^17`

The whole sum becomes

`sqrt((64.2timescancel(10^8))/(25.84times10^(cancel(\ 17)9))`

which is

`sqrt((64.2)/(25.84times10^9))`

or

`sqrt(64.2/25840000000)`

`=0.0000498.......`

or

`4.98.....times10^-5`

**Example 3**

Work out the following giving your answer in standard form

`(5times10^3)/(9times10^-5)`

## Method 1

Actually carry out the division of this sum

`5times10^3=5times10times10times10=5000`

`9times10^-5=9/(10times10times10times10times10)`

`=0.00009`

so the sum becomes

`5000/0.00009`

`=55,555,555.dot5`

In standard form this would be:

remember moving decimal place to the left and `+1` to the power

`5.dot5times10^7`

## Method 2

The alternative way is to add or subtract the indices

`(5times10^3)/(9times10^-5)`

is the same as

`(5times10^3times10^5)/9`

which is

`5/9times10^8`

which is

`0.5dot5times10^8`

but in standard form

remember moving the decimal to the right and `-1` to the power

so

`0.5dot5times10^8`

becomes

`5.dot5times10^7`