Section 10.3 Multiplying and Simplifying Radical Expressions
Product Rule
𝑛
𝑛
 If √𝑎 and √𝑏 are real numbers, then
𝑛
𝑛
𝑛
√𝑎 ∙ √𝑏 = √𝑎𝑏
Simplifying Radical Expressions
 Step 1: Find the index, n, of the radical expression.
 Step 2: Rewrite the radicand as the product of two factors; one of which is the greatest
perfect nth power factor.
 Step 3: Use the product rule and find the nth root of the perfect nth power.
Exercises
(Solution 1)
Step 1: Identify indices
Indices of √3 and √14 are 2.
Step 2: Use product rule, if indices are same.
√3 ∙ √14 = √3 ∙ 14 = √42
Step 3: Find the perfect nth power factor
Since the index is 2, find the perfect square factor.
42 = 2 ∙ 3 ∙ 7. No square factor.
Step 4: Simplifying radicals
The radicand 42 does not include the perfect square
factor. Therefore, √42 is simplified.
(Solution 2)
Step 1: Identify indices
3
3
Indices of √2 and √11 are 3.
Step 2: Use product rule, if indices are same.
3
3
3
√2 ∙ √11 = √2 ∙ 11 = √22
Step 3: Find the perfect nth power factor
Since the index is 3, find the perfect square factor.
22 = 2 ∙ 11. No square factor.
Step 4: Simplifying radicals
The radicand 22 does not include the perfect cube
factor. Therefore, √22 is simplified.
(Solution 3)
Step 1: Identify indices.
Indices of √10𝑥 and �3𝑦 are 2.
Step 2: Use product rule, if indices are same.
√10𝑥 ∙ �3𝑦 = �10𝑥 ∙ 3𝑦 = �30𝑥𝑦
Step 3: Find the perfect nth power factor
Since the index is 2, find the perfect square factor.
30𝑥𝑦 = 2 ∙ 3 ∙ 5 ∙ 𝑥 ∙ 𝑦. No square factor.
Step 3: Simplifying radicals
The radicand 30xy does not include the perfect
square factor. Therefore, �30𝑥𝑦 is simplified.
Cheon-Sig Lee
Page 1
Section 10.3 Multiplying and Simplifying Radical Expressions
(Solution 4)
Step 1: Identify indices. The index is 2.
Step 2: Find the perfect nth power factor.
Since the index is 2, find the perfect square factor.
175 = 5 ∙ 5 ∙ 7 = 52 ∙ 7 = 25 ∙ 7.
The perfect square factor is 25 because 52 = 25
Step 3: Simplifying radicals
√175 = √25 ∙ 7 = √25 ∙ √7 = 5√7
(Solution 5)
Step 1: Identify indices. The index is 2
Step 2: Find the perfect nth power factor.
Since the index is 2, find the perfect square factor.
12𝑥 = 2 ∙ 2 ∙ 3 ∙ 𝑥 = 22 ∙ 3 ∙ 𝑥 = 4 ∙ 3 ∙ 𝑥.
The perfect square factor is 4 because 22 = 4
Step 3: Simplifying radicals
√12𝑥 = √4 ∙ 3𝑥 = √4 ∙ √3𝑥 = 2√3𝑥
(Solution 6)
Step 1: Identify indices. The index is 3
Step 2: Find the perfect nth power factor.
Since the index is 3, find the perfect cube factor.
32𝑥𝑦 3 = 2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 ∙ 𝑥 ∙ 𝑦 3 = 23 ∙ 22 ∙ 𝑥 ∙ 𝑦 3 .
So, −32𝑥𝑦 3 = (−2)3 ∙ 22 ∙ 𝑥 ∙ 𝑦 3 = −8 ∙ 4 ∙ 𝑥 ∙ 𝑦 3
Thus, the perfect cube factor is −8 and 𝑦 3
Step 3: Simplifying radicals
3
3
3
3
3
3
�−32𝑥𝑦 3 = �−8 ∙ 4 ∙ 𝑥 ∙ 𝑦 3 = √−8 ∙ �𝑦 3 ∙ √4𝑥 = −2𝑦 √4𝑥
(Solution 7)
Step 1: Identify index. The index is 2.
Step 2: Find the perfect nth power factor.
Since the index is 2, find the perfect square factor.
𝑥 15 = 𝑥 14+1 = 𝑥 14 ∙ 𝑥
The perfect square factor is 𝑥 14 because 𝑥 14 = (𝑥 7 )2
Step 3: Simplifying radicals
14
�𝑥 15 = �𝑥 14 ∙ 𝑥 = �𝑥 14 ∙ √𝑥 = 𝑥 2 √𝑥 = 𝑥 7 √𝑥
(Solution 8)
Step 1: Identify index. The index is 2
Step 2: Find the perfect nth power factor.
Since the index is 3, find the perfect cube factor.
32𝑥 16 𝑦 6 = 25 𝑥 16 𝑦 6 = 23+2 𝑥 15+1 𝑦 6
= 23 ∙ 22 ∙ 𝑥 15 ∙ 𝑥 ∙ 𝑦 6 = 8 ∙ 4 ∙ 𝑥 15 ∙ 𝑥 ∙ 𝑦 6
Perfect cube factors are 8, 𝑥 15 , and 𝑦 6 .
Step 3: Simplifying radicals
3
3
�32𝑥 16 𝑦 6 = �8 ∙ 4 ∙ 𝑥 15 ∙ 𝑥 ∙ 𝑦 6
3
3
3
3
= √8 ∙ �𝑥 15 ∙ �𝑦 6 ∙ √4𝑥
15
6
3
= 2 ∙ 𝑥 3 ∙ 𝑦 3 ∙ √4𝑥
3
= 2𝑥 5 𝑦 2 √4𝑥
Cheon-Sig Lee
Page 2
Section 10.3 Multiplying and Simplifying Radical Expressions
(Solution 9)
Step 1: Multiply radicals
Since indices are same, radicals can be multiplied.
√15 ∙ √6 = √15 ∙ 6 = √90
Step 2: Find the perfect nth power factor.
Since the index is 2, find the perfect square factor.
90 = 2 ∙ 3 ∙ 3 ∙ 5 = 32 ∙ 2 ∙ 5 = 9 ∙ 10
The perfect square factor is 9 because 32 = 9
Step 3: Simplify radicals
√15 ∙ √6 = √90 = √9 ∙ 10 = √9 ∙ √10 = 3√10
(Solution 10)
Step 1: Multiply radicals
Since indices are same, radicals can be multiplied.
√2𝑥 ∙ �12𝑦 = �2𝑥 ∙ 12𝑦 = �24𝑥𝑦
Step 2: Find the perfect nth power factor.
Since the index is 2, find the perfect square factor.
24𝑥𝑦 = 2 ∙ 2 ∙ 2 ∙ 3 ∙ 𝑥 ∙ 𝑦 = 22 ∙ 2 ∙ 3𝑥𝑦 = 4 ∙ 6𝑥𝑦
The perfect square factor is 4 because 22 = 4
Step 3: Simplify radicals
√2𝑥 ∙ �12𝑦 = �24𝑥𝑦 = �4 ∙ 6𝑥𝑦 = √4 ∙ �6𝑥𝑦 = 2�6𝑥𝑦
(Solution 11)
Step 1: Multiply radicals
Since indices are same, radicals can be multiplied.
�27𝑥𝑦 ∙ �9𝑥𝑦 2 = �27𝑥𝑦 ∙ 9𝑥𝑦 2 = �243𝑥 2 𝑦 3
Step 2: Find the perfect nth power factor.
Since the index is 2, find the perfect square factor.
243𝑥 2 𝑦 3 = 3 ∙ 3 ∙ 3 ∙ 3 ∙ 3 ∙ 𝑥 2 ∙ 𝑦 2+1
= 34 ∙ 3 ∙ 𝑥 2 ∙ 𝑦 2 ∙ 𝑦
= 81 ∙ 3 ∙ 𝑥 2 ∙ 𝑦 2 ∙ 𝑦
Perfect square factors are 81, 𝑥 2 , and 𝑦 2 because
34 = (32 )2 = (9)2 = 81
Step 3: Simplify radicals
�243𝑥 2 𝑦 3 = �81 ∙ 3 ∙ 𝑥 2 ∙ 𝑦 2 ∙ 𝑦
= √81 ∙ �𝑥 2 ∙ �𝑦 2 ∙ �3𝑦
= 9𝑥𝑦�3𝑦
(Solution 12)
Step 1: Multiply radicals
Since indices are same, radicals can be multiplied.
5√3 ∙ 2√24 = 5 ∙ 2 ∙ √3 ∙ √24 = 10√3 ∙ 24 = 10√72
Step 2: Find the perfect nth power factor.
Since the index is 2, find the perfect square factor.
72 = 2 ∙ 2 ∙ 2 ∙ 3 ∙ 3 = 2 ∙ 3 ∙ 2 ∙ 3 ∙ 2 = 62 ∙ 2 = 36 ∙ 2
Perfect square factors are 36 because 62 = 36
Step 3: Simplify radicals
5√3 ∙ 2√24 = 10√72 = 10√36 ∙ 2
= 10√36 ∙ √2 = 10 ∙ 6 ∙ √2
= 60√2
Cheon-Sig Lee
Page 3
Section 10.3 Multiplying and Simplifying Radical Expressions
(Solution 13)
Step 1: Multiply radicals
Since indices are same, radicals can be multiplied.
�6𝑥 6 ∙ �12𝑥 5 = �6𝑥 6 ∙ 12𝑥 5 = �72𝑥 11
Step 2: Find the perfect nth power factor.
Since the index is 2, find the perfect square factor.
72𝑥 11 = 2 ∙ 2 ∙ 2 ∙ 3 ∙ 3 ∙ 𝑥 10+1
= 2 ∙ 3 ∙ 2 ∙ 3 ∙ 2 ∙ 𝑥 10 ∙ 𝑥
= 6 ∙ 6 ∙ 2 ∙ 𝑥 10 ∙ 𝑥
= 36 ∙ 𝑥 10 ∙ 2𝑥
Perfect square factors are 36 and 𝑥 10 because
62 = 36 and (𝑥 5 )2 = 𝑥 10
Step 3: Simplify radicals
�6𝑥 6 ∙ �12𝑥 5 = �72𝑥 11 = �36 ∙ 𝑥 10 ∙ 2𝑥
= √36 ∙ �𝑥 10 ∙ √2𝑥
10
= 6 ∙ 𝑥 2 ∙ √2𝑥
= 6𝑥 5 √2𝑥
(Solution 14)
Write the walking speed of a dinosaur whose leg length is 15feet.
Step 1: Define variables, information, and the question.
W(x) = the walking speed = ?
x = the length of the leg = 15 feet.
The question is finding the walking speed at 15 feet,
another words evaluating W(15)
Step 2: Substituting and evaluating.
𝑊(𝑥) = 4√3𝑥
𝑊(15) = 4√3𝑥 = 4√3 ∙ 15 = 4√45 = 4√9 ∙ 5
= 4 ∙ √9 ∙ √5 = 4 ∙ 3 ∙ √5 = 12√5
Cheon-Sig Lee
Estimate the walking speed at 15 feet
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