Simplifying square roots
When you simplify square roots write down:
|
2 |
= |
`sqrt4` |
|
3 |
= |
`sqrt9` |
|
4 |
= |
`sqrt16` |
|
5 |
= |
`sqrt25` |
|
6 |
= |
`sqrt36` |
|
7 |
= |
`sqrt49` |
|
8 |
= |
`sqrt64` |
|
9 |
= |
`sqrt81` |
|
10 |
= |
`sqrt100` |
Find the biggest square that
divides the number in the root.
Examples
1. Simplify
`sqrt108`
Write down what we know:
`2=sqrt4,` `3=sqrt9,` `4=sqrt16`
`5=sqrt25,\ \ 6=sqrt36,\ \ 7=sqrt49`
The biggest square that divides into 108 is 36
`sqrt108=sqrt(3times36)`
And if we write out what we know from multiplying surds
`sqrt9timessqrt9=3times3=9=sqrt81=sqrt(9times9)`
Therefore `sqrt(3times36)=sqrt3timessqrt36`
`sqrt3timessqrt36=sqrt3times6=6sqrt3`
Answer: `sqrt108=6sqrt3`
2. Simplify
`sqrt45`
Write down what we know
`2=sqrt4,` `3=sqrt9,` `4=sqrt16`
`5=sqrt25,\ \ 6=sqrt36,\ \ 7=sqrt49`
The biggest square that divides 45 is 9
`sqrt45 =sqrt(9times5)`
And if we write down what we know from multiplying surds
`sqrt9timessqrt9=3times3=9`
`=sqrt81=sqrt(9times9)`
Therefore `sqrt45 =sqrt(9times5)=sqrt9timessqrt5`
`sqrt9timessqrt5 =3sqrt5`
Answer: `sqrt45=3sqrt5`
3. Simplify
`sqrt80`
Write down what we know
`2=sqrt4,` `3=sqrt9,` `4=sqrt16`
`5=sqrt25,\ \ 6=sqrt36,\ \ 7=sqrt49`
The biggest square that divides 80 = 4
(It is actually 16 but this example shows you that you can simplify again)
`sqrt80=sqrt(4times20)`
And if we write down what we know from multiplying surds
`sqrt9timessqrt9=3times3=9`
`=sqrt81=sqrt(9times9)`
Therefore `sqrt80=sqrt(4times20)=sqrt4timessqrt20`
`=2timessqrt20`
But `sqrt20` can be simplified
The biggest square that divides 20 = 4
`2timessqrt20=2timessqrt(4times5)`
And if we write down what we know from multiplying surds
`sqrt9timessqrt9=3times3=9`
`=sqrt81=sqrt(9times9)`
Therefore `2timessqrt(4times5)=2timessqrt4timessqrt5`
`=2times2timessqrt5`
`2times2timessqrt5=4sqrt5`
Answer: `sqrt80=4sqrt5`
