MCR 3U1
Unit 3
Lesson 4: Operations with Radicals
The following properties are used to simplify radical expressions:
1.
ab  a  b , a  0 , b  0
2.
a
a
, a  0, b  0

b
b
A radicand is the expression under the radical (square root) sign.
A radical is in simplest form when:
 The radicand has no perfect square factors other than 1.
 The radicand does not contain a fraction.
 No radical appears in the denominator of a fraction.
Example 1: Simplify the following radicals using the perfect-square method.
a)
b)
75
c)
24
33
3
d)
3
16
Note: you can also work “backwards” and express a mixed radical as an entire radical.
Change the mixed radical: 2 3
Example 2: To multiply radical expressions, multiply the whole numbers and multiply the
radicals. Simplify the radical product where possible.
a)
b) 4 3  2 7
2  10
c) 2 5  3 10
You can also express radicals in simplest radical form by adding or subtracting like radicals. For
example:
4 32 3 3
 7 3 1 3
6 3
MCR 3U1
Unit 3
Example 3: Simplify the following expressions.
a) 8 5  3 7  7 7  4 5
b)
c) 5 27  4 48
20  45
To multiply any radical expression, use the distributive property and add or subtract like radicals.
Example 4: Simplify each radical expression.
a)
3 ( 6  1)
b)
(2 3  1)(3 3  2)
Remember that you should never leave a radical in the denominator because it is bad form! You
can easily simplify a radical expression with a radical in the denominator. Just multiply the
numerator and the denominator by this monomial radical.
Example 5: State the following expression in simplified form:
3 6
4 10
.