Difficult examples

1. Simplify

$4/sqrt8$

The first thing you should attempt is to rationalise the denominator.

$4/sqrt8=4/{sqrt8timessqrt8/sqrt8}=4/{{sqrt8timessqrt8}/sqrt8}=4/{{8}/sqrt8}$

So we have rationalised the denominator

i.e.

$4/sqrt8=4/(8/sqrt8)$

But we can go further

$4/{{8}/{sqrt8}}={4timessqrt8}/8=sqrt8/2$

But now we can also simplify square roots

Simplify $sqrt8$

Write down what we know

$2=sqrt4,\ \ \ \ 3=sqrt9,\ \ \ \ 4=sqrt16$

$5=sqrt25,\ \ 6=sqrt36,\ \ 7=sqrt49$

$sqrt8=sqrt(4times2)$

Try something we know using $sqrt9$

$sqrt(9times4)=sqrt36=6$

Is that the same as

$sqrt9timessqrt4=3times2=6$

Yes it is

Therefore $sqrt(4times2)=sqrt4timessqrt2$

$sqrt4timessqrt2=2timessqrt2$

To finish

$sqrt8/2=(cancel2timessqrt2)/cancel(2)=sqrt2$

Answer: $4/sqrt8=sqrt2$

2. Simplify $sqrt(a^3)$

Try something we know using $sqrt9$

We know $sqrt(9^2)$ is the same as

$sqrt9timessqrt9$

i.e. both = 9

Therefore $sqrt(9^3)=sqrt9timessqrt9timessqrt9$

Is the same as

$=sqrt(9^2)timessqrt9=9sqrt9$

Therefore $=sqrt(a^3)$

Is the same as

$=sqrt(a^2)timessqrta=asqrta$

Answer: $sqrt(a^3)=asqrta$

3. Rationalise the denominator

$2/(9sqrt5)$

Rationalise the denominator = Turn the surd of the denominator into a fraction.

$2/{9sqrt5}=2/{{9timessqrt5timessqrt5}/sqrt5}=2/{{9times5}/sqrt5}=2/{45/sqrt5}$

We have rationalised the denominator but we can go further

$2/{{45}/sqrt5}={2sqrt5}/45$

Answer: $2/{9sqrt5}={2sqrt5}/45$

4. Simplify

$(3+sqrt2)(3-sqrt2)$

Multiply out

Multiply this surd example

$=3times3+3times(-sqrt2)+3timessqrt2-sqrt2timessqrt2$

$=9-3sqrt2+3sqrt2 -sqrt(2times2)$

$=9-cancel(3sqrt2)+cancel(3sqrt2) -sqrt4$

$=9-2$

$= 7$

Answer: = 7

5. Simplify

$(x+sqrty)^2$

Multiply out

Give this a go, no numbers needed, lower tier students this is your most testing example

Answer:

$=x^2+xsqrty+xsqrty+sqrtytimessqrty$

$=x^2+2xsqrty+sqrtytimessqrty$

$=x^2+2xsqrty+sqrt(y^2)$

$=x^2+2xsqrty+y$

Answer: $=x^2+2xsqrty+y$

6. Simplify

$(x+sqrty)(x-sqrty)$

Multiply out

Give this a go, no numbers needed, lower tier students this is your hardest example

$=x^2-xsqrty+xsqrty-sqrtytimessqrty$

$=x^2-cancel(xsqrty)+cancel(xsqrty)-sqrt(y^2)$

$=x^2-y$

Answer: $=x^2-y$

7. Simplify

$6sqrt2+sqrt18$

First simplify

$sqrt18$

Write down what we know

$2=sqrt4,\ \ \ \ 3=sqrt9,\ \ \ \ 4=sqrt16$

$5=sqrt25,\ \ 6=sqrt36,\ \ 7=sqrt49$

The biggest square that divides into 18 is 9.

Therefore $sqrt18 = sqrt(2times9)$

Try something we know using $sqrt9$

$sqrt(4times9)=sqrt36=6$

Is that the same as

$sqrt4timessqrt9=3times2=6$

Yes it is

Therefore $sqrt(2times9)=sqrt2timessqrt9$

$sqrt2timessqrt9=3timessqrt2$

So now getting back to the original question

$6sqrt2+sqrt18$

$=6sqrt2+3sqrt2$

Try something we know using $sqrt9$

$6sqrt9+3sqrt9=6times3+3times3$

$=18+9=27$

Is that the same as

$(6+3)sqrt9=9sqrt9=9times3=27$

Yes it is

Therefore $6sqrt2+3sqrt2$

$=(6+3)sqrt2$

$=9sqrt2$

Answer: $6sqrt2+sqrt18=9sqrt2$

8. Simplify

$8sqrt10times5sqrt15$

$8timessqrt10times5timessqrt15$

$8times5timessqrt10timessqrt15$

$40sqrt15timessqrt10$

Try something we know using $sqrt9$

$sqrt9 timessqrt4=3times2=6$

Is that the same as

$sqrt(9 times4)=sqrt36=6$

Yes it is

Therefore $40sqrt15timessqrt10$

$=40timessqrt(15times10)$

$=40sqrt150$

So now simplify

$sqrt150$

Write down what we know

$2=sqrt4,\ \ \ 3=sqrt9,\ \ \ \ \ \ 4=sqrt16$

$5=sqrt25,\ \ 6=sqrt36,\ \ \ 7=sqrt49$

$8=sqrt64,\ \ 9=sqrt81,\ \ 10=sqrt100$

The biggest square that divides into 150 = 25.

Therefore $sqrt150=sqrt(3times25)=sqrt3timessqrt25$

$sqrt3timessqrt25=5sqrt3$

Get back to

$40sqrt150$

This now equals

$40times5sqrt3$

Therefore $200sqrt3$

Answer: $8sqrt10times5sqrt15=200sqrt3$

9. Simplify

$(5sqrt6+4sqrt3)(3sqrt6-2sqrt3)$

First multiply out the brackets

Higher tier students give this a go

$5sqrt6times3sqrt6-2sqrt3times5sqrt6+4sqrt3times3sqrt6-2sqrt3times4sqrt3$

Working on BIDMAS

Stage 1

Work out $5sqrt6times3sqrt6$

Try something we know using $sqrt9$

$5sqrt9times3sqrt9=5times3times3times3$

$=15times9$

Is that the same as

$5times3timessqrt(9times9)=5times3timessqrt81$

$=15times9$

Yes it is

Therefore $5timessqrt6times3timessqrt6$

$=5times3timessqrt6timessqrt6$

$=15sqrt36$

$=15times6$

$=90$

Stage 2

Work out $-2sqrt3times5sqrt6$

Try something we know using $sqrt9$

$-2timessqrt9times5timessqrt4$

$=-2times3times5times2$

$=-6times10=-60$

Is this the same as

$-2times5timessqrt(9times4)$

$=-10timessqrt36$

$=-10times6=-60$

Yes it is

Therefore $-2timessqrt3times5timessqrt6$

$=-2times5timessqrt3timessqrt6$

$=-10timessqrt(3times6)$

$=-10timessqrt18$

But we can simplify

$sqrt18$

Write down what we know

$2=sqrt4$ , $3=sqrt9$ , $4=sqrt16$

Therefore $sqrt18=sqrt(9times2)=sqrt9timessqrt2$

$sqrt18=3timessqrt2$

Going back

Therfore $-10timessqrt18$

$-10times3timessqrt2$

$-30sqrt2$

Stage 3

$4sqrt3times3sqrt6$

As stage 2

$4timessqrt3times3timessqrt6$

$4times3timessqrt3times sqrt6$

$12timessqrt3timessqrt6$

$12timessqrt(3times6$

$12timessqrt18$

We know from stage 2

$sqrt18=3timessqrt2$

Therefore $12timessqrt18=12times3timessqrt2=36sqrt2$

Stage 4

$-2sqrt3times4sqrt3$

As stage 2

$-2timessqrt3times4timessqrt3$

$-2times4timessqrt3timessqrt3$

$-8timessqrt(3times3)$

$-8timessqrt9$

$-8times3$

$-24$

Putting all the stages together:

$90-30sqrt2+36sqrt2-24$

$90+6sqrt2-24$

$90-24+6sqrt2$

$66+6sqrt2$

Answer: $(5sqrt6+4sqrt3)(3sqrt6-2sqrt3)$

$=66+6sqrt2$