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: $5sqrt2times2sqrt3=10sqrt6$