Section 10.3 Multiplying and Simplifying Radical Expressions
Product Rule
 If √ and √ are real numbers, then
√ ∙√
√
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
Indices of √2 and √11 are 3.
Step 2: Use product rule, if indices are same.
√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 √10 and 3 are 2.
Step 2: Use product rule, if indices are same.
10 ∙ 3
30
√10 ∙ 3
Step 3: Find the perfect nth power factor
Since the index is 2, find the perfect square factor.
30
2 ∙ 3 ∙ 5 ∙ ∙ . No square factor.
Step 3: Simplifying radicals
The radicand 30xy does not include the perfect
square factor. Therefore, 30 is simplified.
Cheon-Sig Lee
www.coastalbend.edu/lee
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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 5 ∙ 7 25 ∙ 7.
25
The perfect square factor is 25 because 5
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.
4∙3∙ .
12
2∙2∙3∙
2 ∙3∙
4
The perfect square factor is 4 because 2
Step 3: Simplifying radicals
2√3
√12
√4 ∙ 3
√4 ∙ √3
(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.
2∙2∙2∙2∙2∙ ∙
2 ∙2 ∙ ∙ .
32
2 ∙2 ∙ ∙
8∙4∙ ∙
So, 32
Thus, the perfect cube factor is 8 and
Step 3: Simplifying radicals
32
8∙4∙ ∙
∙ √4
2 √4
√ 8∙
(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.
∙
because
The perfect square factor is
Step 3: Simplifying radicals
∙
∙√
√
√
(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.
2
2
32
∙ ∙
8∙4∙
∙ ∙
2 ∙2 ∙
, and .
Perfect cube factors are 8,
Step 3: Simplifying radicals
32
8∙4∙
∙ ∙
√8 ∙
2∙
2
Cheon-Sig Lee
www.coastalbend.edu/lee
∙
∙ √4
∙ ∙ √4
√4
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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 3 ∙ 2 ∙ 5 9 ∙ 10
9
The perfect square factor is 9 because 3
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.
2 ∙ 12
24
√2 ∙ 12
Step 2: Find the perfect nth power factor.
Since the index is 2, find the perfect square factor.
4∙6
24
2∙2∙2∙3∙ ∙
2 ∙2∙3
4
The perfect square factor is 4 because 2
Step 3: Simplify radicals
24
4∙6
2 6
√2 ∙ 12
√4 ∙ 6
(Solution 11)
Step 1: Multiply radicals
Since indices are same, radicals can be multiplied.
27 ∙ 9
27 ∙ 9
243
Step 2: Find the perfect nth power factor.
Since the index is 2, find the perfect square factor.
3∙3∙3∙3∙3∙ ∙
243
3 ∙3∙ ∙ ∙
∙
81 ∙ 3 ∙ ∙
because
Perfect square factors are 81, , and
3
9
81
3
Step 3: Simplify radicals
243
81 ∙ 3 ∙ ∙ ∙
√81 ∙
3
9
∙
∙ 3
(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 6 ∙ 2 36 ∙ 2
36
Perfect square factors are 36 because 6
Step 3: Simplify radicals
5√3 ∙ 2√24 10√72 10√36 ∙ 2
10√36 ∙ √2 10 ∙ 6 ∙ √2
60√2
Cheon-Sig Lee
www.coastalbend.edu/lee
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Section 10.3 Multiplying and Simplifying Radical Expressions
(Solution 13)
Step 1: Multiply radicals
Since indices are same, radicals can be multiplied.
6 ∙ 12
6 ∙ 12
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∙
∙
6∙6∙2∙
∙2
36 ∙
because
Perfect square factors are 36 and
36 and
6
Step 3: Simplify radicals
72
36 ∙
∙2
6 ∙ 12
∙ √2
√36 ∙
∙ √2
6∙
6 √2
(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.
4√3
4√3 ∙ 15 4√45 4√9 ∙ 5
15
4√3
4 ∙ √9 ∙ √5 4 ∙ 3 ∙ √5 12√5
Cheon-Sig Lee
Estimate the walking speed at 15 feet
www.coastalbend.edu/lee
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