Lesson 5.2: Multiplying and Dividing Radical Expressions
Specific Outcome: Solve problems that involve operations on radicals and radical expressions
with numerical and variable radicands.
Steps for Multiplying Radicals:



Multiply the coefficients and multiply the radicands. You can only multiply radicals if
they have the same index.
If the index is even, state the restrictions on the radicand.
When multiplying radicals with more than one term, use the distributive property and
simplify.
Example 1: Simplify.
𝑎) (3√5)(√10)
𝑏) (4√14𝑥)(2√7𝑥 3 ), 𝑥 ≥ 0
𝑐) √3(−5√10 + √6)
𝑑) (5√2𝑥 + √5)(−4√2𝑥 + √5𝑥), 𝑥 ≥ 0
3
3
Example 2: The expression (7𝑥 √8𝑥𝑦 2 )(3√8𝑥 2 𝑦 2 ) can be simplified into the form 𝑎𝑥 2 𝑦 3√𝑦 ,
the value of 𝑎 is ______.
(Record your answer in the numerical response box from left to right.)
 Conjugate: Two binomial factors whose product is the difference of two squares. To find
the conjugate of a binomial, reverse the sign on the second term in the binomial, creating
the difference of two squares. The result is a rational number.
 e.g.,
binomial: (4 − √6)
Steps for Dividing Radicals:



conjugate: (4 + √6)
Divide the coefficients and divide the radicands.
Always rationalize the denominator when it contains a square root.
If the denominator contains a square root binomial, then
1. Find the conjugate of the denominator
2. Multiply the numerator and the denominator by the conjugate
3. Simplify
Example 3: Simplify.
𝑎)
𝑐)
√15𝑥𝑦
√5𝑥
, 𝑥 ≥ 0, 𝑦 ≥ 0
𝑏)
5
𝑑)
3√2−4
3
8𝑘
𝑒) √ 3
8√5
2√3
7
5√3−√2
* Without simplifying, why are there no restrictions on k here?
Example 4: Simplify
a) −√45 + 2√5 − √20
c)
b) (3 − √2𝑥)(3 + √2𝑥)
3
d)
√5+4
3+4√3
√2+2√5
Example 5: Given the following diagram, the volume of the right rectangular prism can be
expressed in the form 𝑎√𝑏 , where 𝑎 𝑎𝑛𝑑 𝑏 𝜖 ℵ. The value of 𝑎 ÷ 𝑏 is _______.
(Record your answer in the numerical response box from left to right.)
6√2 − 2√3
√10
6√2 + 2√3
√𝑚−2
)
√𝑚
Example 6: Anthony simplified the expression (6−
as shown. Identify any errors he made,
including restrictions he has identified. Show the correct simplification and restriction(s) on the
variable.
√𝑚−2
)
√𝑚
(6−
6+ √𝑚
√𝑚−2
) (6+ 𝑚)
√𝑚
√
= (6−
=
6√𝑚−12+𝑚
6−𝑚
= 6√𝑚 − 2, where m > 0
Practice Questions: Page 289 # 1 – 6(a, c’s), 8(c), 10 (a, d), 13, 25(a, b, c)