8.2 – Notes
Math 51 – Prof. Beydler
Page 1 of 4
Multiplying, Dividing, and Simplifying Radicals
Product Rule for Radicals
and
√𝒂𝒃 = √𝒂 ⋅ √𝒃
√𝒂 ⋅ √𝒃 = √𝒂𝒃
(Note: only true when 𝑎 ≥ 0 and 𝑏 ≥ 0)
Ex 1.
√7 ⋅ √5 =
√11 ⋅ √11 =
√7 ⋅ √𝑘 =
Simplifying Radicals
4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, … are _______________________ (since 4 = 22 , 9 = 32 , …)
8, 27, 64, 125, 216, 343, … are _______________________ (since 8 = 23 , 27 = 33 , …)
16, 81, 256, 625, … are _______________________________ (since 16 = 24 , 81 = 34 , …)
Ex 2.
Simplify:
√80 =
√72 =
Ex 3.
Multiply and simplify.
√6 ⋅ √30 =
3√5 ⋅ 4√10 =
√8 ⋅ √12 =
8.2 – Notes
Math 51
Quotient Rule for Radicals
𝑎 √𝑎
𝑎
√𝑎
√ =
and
=√
𝑏 √𝑏
𝑏
√𝑏
(Note: only true when 𝑎 ≥ 0 and 𝑏 ≥ 0)
Ex 4.
Simplify.
25
√
9
√288
√2
=
=
3
√ =
4
8√50
4√5
=
3
7
√8 ⋅ √2 =
Ex 5.
Simplify. Assume that all variables represent nonnegative real numbers.
√16𝑥 8 =
√𝑥 5 =
√
5
𝑥2
=
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8.2 – Notes
Math 51
𝑛
𝑛
𝑛
Note: √𝑎 ⋅ √𝑏 = √𝑎𝑏
Ex 6.
Simplify.
3
√40 =
4
√32 =
27
3
√125 =
3
√𝑥 6 =
3
√27𝑥 12 =
3
√32𝑥 4 =
3
√
𝑥 15
1000
=
𝑛
and
√𝑎
√𝑏
𝑛
𝑛
𝑎
= √𝑏
Page 3 of 4
8.2 – Notes
Math 51
Page 4 of 4
Practice
1. Simplify.
a) √5 ⋅ √6
b) √64
c) 6√40
d) √50 ⋅ √72
e)
√200
√2
14
f) √72
g) √50𝑥 20
3
h) √192
4
i) √243
3
j) √125𝑥 15
3
k) √24𝑥 4
Q: April says May is a liar. May says June is a liar. June says April and May are both liars. If only one
person is telling the truth, who is it?