Mammoth Memory

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`

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