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By the end of this section, you will be able to:
  • Multiply square roots
  • Use polynomial multiplication to multiply square roots

Before you get started, take this readiness quiz.

  1. Simplify: ( 3 u ) ( 8 v ) .
    If you missed this problem, review [link] .
  2. Simplify: 6 ( 12 7 n ) .
    If you missed this problem, review [link] .
  3. Simplify: ( 2 + a ) ( 4 a ) .
    If you missed this problem, review [link] .

Multiply square roots

We have used the Product Property of Square Roots to simplify square roots by removing the perfect square factors. The Product Property of Square Roots says

a b = a · b

We can use the Product Property of Square Roots ‘in reverse’ to multiply square roots.

a · b = a b

Remember, we assume all variables are greater than or equal to zero.

We will rewrite the Product Property of Square Roots so we see both ways together.

Product property of square roots

If a , b are nonnegative real numbers, then

a b = a · b and a · b = a b

So we can multiply 3 · 5 in this way:

3 · 5 3 · 5 15

Sometimes the product gives us a perfect square:

2 · 8 2 · 8 16 4

Even when the product is not a perfect square, we must look for perfect-square factors and simplify the radical whenever possible.

Multiplying radicals with coefficients is much like multiplying variables with coefficients. To multiply 4 x · 3 y we multiply the coefficients together and then the variables. The result is 12 x y . Keep this in mind as you do these examples.

Simplify: 2 · 6 ( 4 3 ) ( 2 12 ) .

Solution


  1. 2 · 6 Multiply using the Product Property. 12 Simplify the radical. 4 · 3 Simplify. 2 3


  2. ( 4 3 ) ( 2 12 ) Multiply using the Product Property. 8 36 Simplify the radical. 8 · 6 Simplify. 48

Notice that in (b) we multiplied the coefficients and multiplied the radicals. Also, we did not simplify 12 . We waited to get the product and then simplified.

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Simplify: 3 · 6 ( 2 6 ) ( 3 12 ) .

3 2 36 2

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Simplify: 5 · 10 ( 6 3 ) ( 5 6 ) .

5 2 90 2

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Simplify: ( 6 2 ) ( 3 10 ) .

Solution

( 6 2 ) ( 3 10 ) Multiply using the Product Property. 18 20 Simplify the radical. 18 4 · 5 Simplify. 18 · 2 · 5 36 5

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Simplify: ( 3 2 ) ( 2 30 ) .

12 15

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Simplify: ( 3 3 ) ( 3 6 ) .

27 2

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When we have to multiply square roots, we first find the product and then remove any perfect square factors.

Simplify: ( 8 x 3 ) ( 3 x ) ( 20 y 2 ) ( 5 y 3 ) .

Solution


  1. ( 8 x 3 ) ( 3 x ) Multiply using the Product Property. 24 x 4 Simplify the radical. 4 x 4 · 6 Simplify. 2 x 2 6


  2. ( 20 y 2 ) ( 5 y 3 ) Multiply using the Product Property. 100 y 5 Simplify the radical. 10 y 2 y
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Simplify: ( 6 x 3 ) ( 3 x ) ( 2 y 3 ) ( 50 y 2 ) .

3 x 2 2 10 y 2 y

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Simplify: ( 6 x 5 ) ( 2 x ) ( 12 y 2 ) ( 3 y 5 ) .

2 x 3 3 6 y 2 y

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Simplify: ( 10 6 p 3 ) ( 3 18 p ) .

Solution

( 10 6 p 3 ) ( 3 18 p ) Multiply. 30 108 p 4 Simplify the radical. 30 36 p 4 · 3 30 · 6 p 2 · 3 180 p 2 3

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Simplify: ( 6 2 x 2 ) ( 8 45 x 4 ) .

144 x 3 10

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Simplify: ( 2 6 y 4 ) ( 12 30 y ) .

144 y 2 5 y

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Simplify: ( 2 ) 2 ( 11 ) 2 .

Solution


( 2 ) 2 Rewrite as a product. ( 2 ) ( 2 ) Multiply. 4 Simplify. 2


( 11 ) 2 Rewrite as a product. ( 11 ) ( 11 ) Multiply. 121 Simplify. 11

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Simplify: ( 12 ) 2 ( 15 ) 2 .

12 15

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Simplify: ( 16 ) 2 ( 20 ) 2 .

16 20

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The results of the previous example lead us to this property.

Squaring a square root

If a is a nonnegative real number, then

( a ) 2 = a

By realizing that squaring and taking a square root are ‘opposite’ operations, we can simplify ( 2 ) 2 and get 2 right away. When we multiply the two like square roots    in part (a) of the next example, it is the same as squaring.

Simplify: ( 2 3 ) ( 8 3 ) ( 3 6 ) 2 .

Solution


( 2 3 ) ( 8 3 ) Multiply. Remember, ( 3 ) 2 = 3 . 16 · 3 Simplify. 48


( 3 6 ) 2 Multiply. 9 · 6 Simplify. 54

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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