 # 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