Simplifying Square Root
Expressions
Numbers with a Root
Radical numbers are typically irrational numbers
(unless they simplify to a rational number). Our
calculator gives:
2 1.41421
But the decimal will go on forever and not repeat
because it is an irrational number. For the exact
answer just use:
2
Some radicals can be simplified similar to
simplifying a fraction.
Radical Product Property
a b
ab
ONLY when a≥0 and b≥0
For Example:
9  16  9 16  144  12
9  16  3  4  12
Equal
Perfect Squares
The square of whole
numbers.
1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100 ,
121, 144 , 169 , 196 , 225, etc
Simplifying Square Roots
1. Check if the square root is a whole number
2. Find the biggest perfect square (4, 9, 16, 25,
36, 49, 64) that divides the number in the root
3. Rewrite the number in the root as a product
4. Simplify by taking the square root of the
perfect square and putting it outside the root
5. CHECK!
Note: A square root can not be simplified if there
is no perfect square that divides it. Just leave it
alone.
ex: √15 , √21, and √17
Simplifying Square Roots
Write the following as a radical (square root) in simplest
form:
36 is the biggest perfect square that divides 72.
Simplify.
72  36  2  36 2  6 2
Rewrite the square root as a product of roots.
27  9  3  9 3  3 3
Ignore the 5 multiplication until the end.
5 32
 5 16  2  5 16 2  5  4  2
 20 2
Simplifying Square Roots
Simplify these radicals:
A) 16
4
B) 8
C) 7
E )4 63  12
2 2
D) 75
7
5 3
F ) 128  8
2
Adding and Subtracting Radicals
Simplify the expressions:
a. 2 3  2  4 3
2 3 4 3  2
2 3  2
Treat the square roots as
variables, then combine like
terms ONLY.
Always simplify a radical first.
b. 4 2  18
4 2  92
4 2 9 2
4 2 3 2
7 2
Multiplication and Radicals
Simplify the expression:
Use the Commutative Property
to Rewrite the expression.
7 10  4 15
7  4  10  15
Simplify and use the Radical
Product Property Backwards.
If possible, simplify more.
Conclusion: Multiply the
numbers outside of the square
root, then multiply the numbers
inside of the square root. Then
simplify.
28 10 15
28 150
28 25 6
28  5 6
140 6
Distribution and Radicals
Rewrite the expression:
5
3√6

6 4 3 3 6 2 3
-2√3
-10√18
15√36
5√6
90 -30√2
12√18 -8√9
4√3
36√2 -24

Find the Sum.
90  30 2  36 2  24
66  6 2
Remember: Multiply the numbers outside of
the square root, then multiply the numbers
inside of the square root. Then simplify.
Combine like terms.
Fractions and Radicals
Simplify the expressions:
a.
5 7
10
b.
There is nothing to
simplify because the
square root is
simplified and every
term in the fraction
can not be divided by
10.
Make sure to
simplify the
fraction.
4  12
2
4 4 3
2
4 2 3
2
2 2 3


2
2 3
c.
15 180
9
15 36 5
9
15 6 5
9
3 5 2 5

33
5 2 5
3

Radical Quotient Property
a

b
a
b
ONLY when a≥0 and b≥0
For Example:
64
16
64
16

64
16

 4 2
8
4
2
Equal
The Square Root of a Fraction
Write the following as a radical (square root) in simplest
form:
Take the square root of the numerator and the denominator
3
3
3


2
4
4
Simplify.
Rationalizing a Denominator
The denominator of a fraction can not contain a radical. To
rationalize the denominator (rewriting a fraction so the
bottom is a rational number) multiply by the same
radical.
Simplify the following expressions:
5 2
5 2
5
2



2
2
2 2
2
 
6 3 6 3 3 2 3 2 3
6 3
3






2
35
5
15
53
3 5 3
5 3
6
 
WARNING
In general:
ab 
a
b
For Example:
9  16  25  5
9  16  3  4  7
Not
Equal