Radical Expressions: Simplifying
Radicals
A radical expression consists of the following parts



Index – indicating the degree of the root
Radical symbol
Radicand – number under the radical sign
The Mechanics: Simplifying Radicals
A radical expression is simplified when
1.
The radicand has no perfect-square factors
other than 1
2.
The radicand has no fractions
3.
No denominator contains a radical
Identifying the violations
Determine which condition is violated for the following radicals.
1. √48
2. √27n2
13
3.
36
6. √12
7. √32
8.
2
6
4.
5
2
5.
9.
96
12
10.
6
5
27
3
Simplifying Radicals: C1 - Removing Perfect Square Factors
According to condition 1, the radicand should not have any perfect-square factors other than 1. If it does,
then
 Write the radicand as a product of prime factors.
 Identify any perfect squares to be removed.
 Remove perfect squares, leaving non-perfect squares as radicands.
Examples:
11. √48
Step 1: Factor out
12. √200
Step 1: Factor out
Step 2: Look for pairs
2 pairs of 2 therefore 2*2 is what comes out and 3
remains in.
Step 2: Look for pairs
1 pair of 10s therefore 10 is what comes out and 2
remains in.
Step 3: Write out in simplest form
±4√3
Step 3: Write out in simplest form
±10√2
Practice: Removing Perfect Square Roots
13. √27
14. √50
The Mechanics: Simplifying Radicals
15.
1
16. √243
Rev C
Radical Expressions: Simplifying
Simplifying Radicals: C1b – Removing Variable Factors


As with numbers, variables are treated the same way.
Shortcut for square roots:
o if variable is even power, then pull variable out and raise it to ½ the power
o if variable is odd power, then pull variable out & raise it to (n-1)/2 power & leave variable to
1st power in radicand
Examples: Remaining variable factors
17.
Step 1: Identify your power and index.
Power = 7
Index = 2
18. a 6
Step 1: Identify your power and index.
Power of a = 6
Index = 2
Step 2: Determine what comes out & what stays in.
= 3remainder 1
Out:
In:
Step 3: Write out in simplest form.
Step 2: Determine what comes out & what stays in.
= 3 remainder 0
Out:
In: nothing
Step 3: Write out in simplest form.
Practice: Removing Variable Factors
19. √16a3
20.
21. √28a7
22. √75x5
Simplifying Radicals: C1c – Multiplying Radicals
When 2 radical expressions are multiplied together, either
 As 1 radicand, write each radicand as a product of primes
Or
 Use the multiplication property of radicals
In either case, follow previously state guidelines to simplify under the radical.
Examples: Multiplying Radicals
Method 1
√8 * √12 =
Method 2
3√2b * 4√10b = 12
2*2
12*2b
=4
The Mechanics: Simplifying Radicals
2
= 24b
Rev C
Radical Expressions: Simplifying
Practice: Multiplying Radicals
23. √13 * √52
24. √12 * √32
26. 2√5x2 * 6√10x3
25. 5√3c * √6c
Simplifying Radicals: C2 – Removing Fractions within Radicals
According to condition 2, the radicand should not have any fractions. If it does, then remove the fractions
from the radicand by
a
a
 Using division property of radicals
=
b
b
Or
 Dividing
Examples (Using Division Property):
16
=
25
a.
=
16
25
Use division property of radicals
4
5
b.
13
=
36
Simplify
=
13
36
13
6
Practice: Removing Fractions within Radicals using Div Prop
27.
11
49
28.
144
9
13
64
29.
30.
49
x4
31.
25 p 3
q2
Examples (Dividing):
32.
88
= √8
11
Divide
12a 3
27a
33.
=
2√2
Simplify
4a 2
9
4a 2
=
9
2a
=
3
The Mechanics: Simplifying Radicals
3
Divide numerator & denominator by 3a.
Use Division Prop of radicals
Simplify
Rev C
Radical Expressions: Simplifying
Practice: Removing Fractions within Radicals by Dividing
34.
90
5
35.
48
75
36.
27 x 3
3x
37.
75x 5
48 x
Simplifying Radicals: C3 – Removing Radicals from Denominator
According to condition 3, the denominator of a fraction should have no radical. If it does, then rationalize
the denominator by
number
 Multiplying both numerator & denominator by
such that the denominator would become a
number
deno min ator
perfect square, cube, etc., (Hint: usually
)
deno min ator
 Use multiplication property of radicals
 Simplify
Examples:
2
5
2
*
5
2 5
=
25
2 5
=
5
=
5
5
Multiply by
5
to make denominator perfect square
5
Use Multiplication Property of Square Roots
Simplify
Practice: Rationalizing the Denominator
38.
43.
3
7
39.
2
6
44.
3
3
40.
5
18t
96
12
45.
6
5
The Mechanics: Simplifying Radicals
4
41.
7m
10
42.
12 5
46.
7
11
12t 3
3 7
47.
12t
Rev C