How to remember

Think of `sqrt9`

If you see surds think 9

“That’s absurd!” (surd) Surely you can remove a square beetroot.” But they couldn’t remove it (cant remove). “We need more lines, perhaps nine.”

Surd examples

1. Add these surds

`sqrt7+4\sqrt7`

This is really asking us to add `1sqrt7+4sqrt7`

Try something we know in `sqrt9`

`sqrt9+4sqrt9=3+4times3=15`

Is that the same as

`5sqrt9=15`

Yes it is

Therefore `1sqrt7+4sqrt7`

`=5sqrt7`

2. Subtract these surds

`5sqrt2-3sqrt2`

Try something we know in `sqrt9`

`5sqrt9 -3sqrt9`

`=5times3-3times3=15-9=6`

Is that the same as

`(5-3)sqrt9=6`

Yes it is

Therefore `5sqrt2-3sqrt2`

`= (5-3)sqrt2`

`= 2sqrt2`

3. Add

`4 sqrt2+3sqrt3`

These cannot be added as these surds are not the same.

It’s not in our rules.

4. Work out

`2sqrt2+3sqrt2-sqrt2`

Try something we know in `sqrt9`

`2sqrt9+3sqrt9-1sqrt9`

`=2times3+3times3-3`

`=6+9-3=12`

Is that the same as

`(2+3-1)sqrt9=12`

Yes it is

Therefore `2sqrt2+3sqrt2-1sqrt2`

`=(2+3-1)sqrt2`

`=4sqrt2`

5. Simplify

`sqrt8/sqrt2`

Start with what we know

`sqrt9/sqrt9=3/3=1=sqrt(9/9)`

Therefore `sqrt8/sqrt2=sqrt(8/2)=sqrt4=2`

Answer: `sqrt8/sqrt2=2`

6. Simplify

`sqrt2timessqrt3`

Try something we know in `sqrt9`

`sqrt4timessqrt9=2times3=6`

Is this the same as

`sqrt{4times9}= sqrt36=6`

Yes it is

Therefore`sqrt2timessqrt3=sqrt{2times3}=sqrt6`

Answer: `sqrt2timessqrt3=sqrt6`

7. Simplify

`sqrt12`

Start with what we know

`sqrt9timessqrt9=3times3=9`

`=sqrt81=sqrt{9times9}`

With this example we can see

`sqrt12=sqrt{3times4}=sqrt3 timessqrt4`

`sqrt3timessqrt4=sqrt3times2=2sqrt{3`

Answer: `sqrt12=2sqrt3`

8. Simplify

`sqrt15/sqrt5`

Write down what we know

`sqrt9/sqrt9=3/3=1=sqrt(9/9)`

Therefore `sqrt9/sqrt9=sqrt(9/9)`

Therfore `sqrt15/sqrt5=sqrt(15/5)=sqrt3`

Answer: `sqrt15/sqrt5=sqrt3`

9. Simplify

`2sqrt3+sqrt2-4sqrt3`

Rearrange

`2sqrt3-4sqrt3+sqrt2`

Try something we know

`2timessqrt9-4sqrt9`

`=2times3-4times3=6-12=-6`

Is that the same as

`(2-4)sqrt9=-2times3=-6`

Yes it is

Therefore `2sqrt3-4sqrt3+sqrt2`

`=(2-4)sqrt3+sqrt2`

`=-2sqrt3+sqrt2`

Answer: `2sqrt3+sqrt2-4sqrt3=-2sqrt3+sqrt2`

10. Simplify

`5sqrt2times2sqrt3`

Try something we know in `sqrt9`

`5sqrt9times2sqrt9=5times3times2times3`

`=15times6=90`

Is this the same as

`5times2sqrt(9times9)=10timessqrt81`

`=10times9=90`

Yes it is

Therefore `5timessqrt2times2timessqrt3`

`= 5times2timessqrt2timessqrt3`

`= 10timessqrt2timessqrt3`

`= 10timessqrt(2times3)`

`= 10timessqrt6`

Answer: `5sqrt2times2sqrt3=10sqrt6`