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`
