Name:
Ms. D’Amato
Date:
Block:
Simplifying Square Roots
Perfect Squares: A perfect square is:
The opposite of squaring a number is:
Perfect squares:
1=1•1
4=2•2
9=3•3
16 = ____ •____
25 = ____ •____
36 = ____ •____
49 = ____ •____
64 = ____ •____
81 = ____ •____
100 = ____ •____
121 = ____ •____
144 = ____ •____
Based on this information, we know that:
4  ____
9  ____
16  ____
25  ____
36  ____
49  ____
64  ____
81  ____
100  ____
A few questions first, to better our understanding! State whether or not the following
numbers are perfect squares. If so, state why.
1.
16
2.
20
3.
-16
4.
45
5.
225
6.
121
Simplifying Radicals Using the Product Property
Product Property of Radicals
ab  a •
b
Why do we care?? What if the value is NOT a perfect square? You are unable to leave this
value as a decimal. This property helps us simplify radical expressions . . .
Steps for simplifying radicals:
1. Make a list of perfect squares and keep it in sight!!!
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
2. Find the largest perfect square which will divide evenly into the number under
your radical sign. (This means when you divide, you get no remainders, no decimals
and no fractions):
Simplify: √48
3. Write the number appearing under your radical sign (the radicand) as the
product (multiplication) of the perfect square and your answer from dividing:
4. Simplify:
8 =
4 •
50 =
2
20 =
What happens if you don’t choose the largest perfect square to start simplifying
your radical??
72
=
9
•
72
8
=
36
•
2
**Always make sure your radical is completely simplified! A radical expression is
completely simplified when the number underneath the radical cannot be divided
evenly by any perfect square other than 1.**
You try:
300
40
75
A little harder!! Let’s try to reduce these radicals together!
4 25
3√50
−2√18
You try:
4 8
−3 4
−3 27
How do you simplify radicals with variables?? Look at these examples and try to find the
pattern:
1.)
√49𝑥 2
2.)
√𝑥 7
3.)
√9𝑥 36
4.)
√4𝑥 9
5.)
√64𝑥 32 𝑦 8
6.)
√36𝑥 11 𝑦13
Solving Quadratic Equations Using Square Roots:
1.)
c2 – 25 = 0
2.)
5w2 – 12 = 8
3.)
2x2 + 11 = 11
Simplifying Cube Roots
A
times.
of a number is the value when a number is multiplied by itself three
What is a perfect cube? It is a number that can be written as the cube (raised to the third
power) of another number.






1x1x1=
2x2x2=
3x3x3=
4x4x4=
5x5x5=
And so on…
Examples:
1.)
3
√64
2.)
3
√−125
3.)
3
√216
4.)
3
5.)
3
6.)
3
7.)
3
8.)
3
√8𝑎3
√27𝑚12
√−64𝑦16
√729
√343𝑥 7 𝑦13