Multiply Square Roots
The product rule of radicals, which we have already been using, can be
generalized as follows:
𝒂√𝒃 ∙ 𝒄√𝒅 = 𝒂𝒄√𝒃𝒅
Another way of stating this rule is we are allowed to multiply the factors
outside the radicals and multiply the factors inside the radicals.
Example 1: −5√14 ∙ 4√6
−5 ∙ 6√14 ∙ 6
Multiply the coefficients, multiply the radicands
−30√84
−30√2 ∙ 2 ∙ 3 ∙ 7
−30 ∙ 2√3 ∙ 7
−60√21
Factor the radicand
Simplify the square root
Multiply the radicands
Notice we could have eliminated a step by factoring the radicands and
simplifying the square root rather than multiplying and then factoring:
−5 ∙ 6√14 ∙ 6
−30√2 ∙ 2 ∙ 3 ∙ 7
−30 ∙ 2√3 ∙ 7
−60√21
Factor the radicand
Simplify the square root
Multiply the radicands
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 2: 7√6(√10 − 5√15)
7√6 ∙ √10 − 7√6 ∙ 5√15
7√2 ∙ 2 ∙ 3 ∙ 5 − 7 ∙ 5√2 ∙ 3 ∙ 3 ∙ 5
7 ∙ 2√3 ∙ 5 − 7 ∙ 5 ∙ 3√2 ∙ 5
Distribute 7√6
Factor the radicands
Multiply
14√15 − 105√10
Example 3: (5 − √2)(3 + 5√2)
5(3 + 5√2) − √2(3 + 5√2)
5 ∙ 3 + 5 ∙ 5√2 − √2 ∙ 3 + √2 ∙ 5√2
15 + 25√2 − 3√2 + 5√4
15 + 25√2 − 3√2 + 10
(15 + 10) + (25√2 − 3√2)
Use the Distributive Property
Distribute 5 and √2
Multiply
Simplify √4 and multiply by 5
Combine like terms
25 + 22√2
Example 4: (√7 + √11)
2
(√7 + √11)(√7 + √11)
√7(√7 + √11) + √11(√7 + √11)
√7 ∙ √7 + √7 ∙ √11 +
√49 + √77 +
7 + √77 +
(7 + 11) +
Write 2 factors of (√7 + √11)
Use the Distributive Property
√11 ∙ √7 + √11 ∙ √11 Distribute √7 and √11
Multiply
√77 + √121
Simplify √49 and √121
√77 + 11
Combine like terms
(√77 + √77)
18 + 2√77
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 5: (8√2 + 4√5)(8√2 − 4√5)
8√2(8√2 − 4√5) + 4√5(8√2 − 4√5)
Use the Distributive Property
8√2 ∙ 8√2 − 8√2 ∙ 4√5 + 4√5 ∙ 8√2 − 4√5 ∙ 4√5 Distribute 8√2 and 4√5
64√4 − 32√10 + 32√10 − 16√25
Multiply
128 − 32√10 + 32√10 − 80
Simplify √4 and √25 and multiply
(128 − 80) + (−32√10 + 32√10 )
Combine like terms
48 + 0
48
The above example 5 illustrates multiplying conjugates. Conjugates are
binomials made up of the same terms with one being an addition and the
other being a subtraction. Notice that whenever we multiply conjugates
the two middle terms have a sum of zero, resulting in a product
containing no square root.
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)