Simplifying Square Roots
*If the radicand in a square root is not a perfect square,
you may be able to simplify the radical by finding the
greatest perfect-square factor of the radicand.
Ex. 1
Simplify
Strategy- The radicand 72 is not a perfect square, so look
for the greatest perfect-square factor of 72.
72 = 4 ∙ 18, so 4 is a perfect-square factor of 72
72 = 9 ∙ 8, so 9 is a perfect-square factor of 72
72 = 36 ∙ 2, so 36 is a perfect-square factor of 72
4, 9, and 36 are all perfect – square factors of 72.
The greatest perfect-square factor of 72 is 36.
=
∙
=6
Solution: The simplified form of
is 6
.
Example 2
Simplify:
Strategy: Simplify the numerator and denominator
=
Factors of 80: {1,2,4, 5, 8,10, 16, 20, 40, 80} *Notice the
greatest perfect square factor of 80 is 16, and 16 ∙ 5=80.
∙
=
=4
4
So,
=
Solution:
=
=
*Note- if you noticed that 16 is a factor of 80, you could
have solved the problem by dividing 80 by 16:
=
Coached Example 1:
Simplify
Think it Through!
The radicand, 63 is not a perfect square.
The greatest perfect –square factor of 63 is 9.
Factor
=
∙
The square root of the perfect-square factor is 3.
So,
=3
Coached Example 2:
Simplify
Think it through!
Simplify the numerator and the denominator.
=
The greatest perfect square factor of 50 is 25.
Factor
=
∙
=5
The square root of the perfect square factor is 5
=9
So,
=
Solution:
=