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.
Simplifying Square Roots
1.
2.
3.
4.
5.
Check if the square root is a whole
number
Find the biggest perfect square (4, 9, 16,
25, 36, 49, 64) that divides the number in
the root
Rewrite the number in the root as a
product
Simplify by taking the square root of the
perfect square and putting it outside the
root
CHECK!
4
=2
16
=4
25
=5
100
= 10
144 = 12
Simplify
Objective - To simplify rational and irrational square roots.
List of Perfect Squares
0= 0
1=1
4= 2
9=3
16 = 4
25 = 5
36  6
49  7
64  8
81  9
100 10
121  11
144  12
169  13
196  14
225  15
400  20
625  25
256  16
289  17
324  18
361  19
For all real values of n, n ³ 0.
Objective - To simplify irrational roots.
a · b = ab
Example:
4 · 9 = 36
2 ·3= 6
ab = a · b
20 = 4 · 5 =
Check using a
calculator!
4· 5
2· 5= 2 5
Simplify.
18 = 9 · 2 = 9 · 2
18 = 6 · 3
3· 2= 3 2
To simplify the number by hand
must have a perfect square factor.
35 =
35
Can’t be
simplified
35
1 35
5 7
No perfect
square factors
Simplify each irrational root.
1) 50 =
4) 12 =
2) 27 =
5)
45 = 9· 5
= 9· 5
=3 5
3) 98 = 49 · 2
6)
28 = 4 · 7
= 4· 7
=2 7
25· 2
= 25 · 2
=5 2
9· 3
= 9· 3
=3 3
= 49 · 2
=7 2
4·3
= 4· 3
=2 3
Simplify.
72
72
72
36· 2
9 ·8
36 · 2
9· 8
6 2
3 8
3 4 ·2
Perfect square
3 4· 2
3·2 · 2 = 6 2
Simplify each irrational root.
1) 32
4)
30
2) 200
5)
48
3) 80
6) -5 60
Ex. 1 Simplify
40
75
32
Ex. 2 Simplify
450
50
80
Homework Day 1
Multiply Radicals
*
To multiply radicals: multiply
everything, then simplify.
Multiplication and Radicals
Simplify the expression:
Use the Commutative Property
to Rewrite the expression.
Simplify and use the Radical
Product Property Backwards.
7 10  4 15
7  4  10  15
28 10 15
28 150
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 25 6
28  5 6
140 6
Ex. 6
Multiply, then simplify.
5 * 35 
2 5 * 4 20 
Homework Day 2
Dividing Radicals
To divide radicals: reduce or divide
what you can and rationalize the
denominator. A square root
CANNOT be in the denominator.
Ex. 8
56

7
5

10
6

7
Reduce or divide,
then rationalize.
Worksheet Day 3