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Simplify: ( 6 11 ) ( 5 11 ) ( 5 8 ) 2 .

330 200

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Simplify: ( 3 7 ) ( 10 7 ) ( −4 6 ) 2 .

210 96

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Use polynomial multiplication to multiply square roots

In the next few examples, we will use the Distributive Property to multiply expressions with square roots.

We will first distribute and then simplify the square roots when possible.

Simplify: 3 ( 5 2 ) 2 ( 4 10 ) .

Solution


3 ( 5 2 ) Distribute. 15 3 2


2 ( 4 10 ) Distribute. 4 2 20 4 2 2 5

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Simplify: 2 ( 3 5 ) 3 ( 2 18 ) .

6 2 5 2 3 3 6

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Simplify: 6 ( 2 + 6 ) 7 ( 1 + 14 ) .

12 + 6 6 7 + 7 2

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Simplify: 5 ( 7 + 2 5 ) 6 ( 2 + 18 ) .

Solution


5 ( 7 + 2 5 ) Multiply. 7 5 + 2 · 5 Simplify. 7 5 + 10 10 + 7 5


6 ( 2 + 18 ) Multiply. 12 + 108 Simplify. 4 · 3 + 36 · 3 2 3 + 6 3 Combine like radicals. 8 3

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Simplify: 6 ( 1 + 3 6 ) 12 ( 3 + 24 ) .

18 + 6 6 + 12 2

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Simplify: 8 ( 2 5 8 ) 14 ( 2 + 42 ) .

−40 + 4 2 2 7 + 14 3

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When we worked with polynomials, we multiplied binomials by binomials. Remember, this gave us four products before we combined any like terms. To be sure to get all four products, we organized our work—usually by the FOIL method.

Simplify: ( 2 + 3 ) ( 4 3 ) .

Solution

( 2 + 3 ) ( 4 3 ) Multiply. 8 2 3 + 4 3 3 Combine like terms. 5 + 2 3

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Simplify: ( 1 + 6 ) ( 3 6 ) .

−3 + 2 6

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Simplify: ( 4 10 ) ( 2 + 10 ) .

−2 + 2 10

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Simplify: ( 3 2 7 ) ( 4 2 7 ) .

Solution

( 3 2 7 ) ( 4 2 7 ) Multiply. 12 6 7 8 7 + 4 · 7 Simplify. 12 6 7 8 7 + 28 Combine like terms. 40 14 7

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Simplify: ( 6 3 7 ) ( 3 + 4 7 ) .

−66 + 15 7

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Simplify: ( 2 3 11 ) ( 4 11 ) .

41 + 14 11

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Simplify: ( 3 2 5 ) ( 2 + 4 5 ) .

Solution

( 3 2 5 ) ( 2 + 4 5 ) Multiply. 3 · 2 + 12 10 10 4 · 5 Simplify. 6 + 12 10 10 20 Combine like terms. −14 + 11 10

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Simplify: ( 5 3 7 ) ( 3 + 2 7 ) .

1 + 9 21

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

−12 20 3

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Simplify: ( 4 2 x ) ( 1 + 3 x ) .

Solution

( 4 2 x ) ( 1 + 3 x ) Multiply. 4 + 12 x 2 x 6 x Combine like terms. 4 + 10 x 6 x

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Simplify: ( 6 5 m ) ( 2 + 3 m ) .

12 8 m 15 m

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Simplify: ( 10 + 3 n ) ( 1 5 n ) .

10 47 n 15 n

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Note that some special products made our work easier when we multiplied binomials earlier. This is true when we multiply square roots, too. The special product formulas we used are shown below.

Special product formulas

Binomial Squares Product of Conjugates ( a + b ) 2 = a 2 + 2 a b + b 2 ( a b ) ( a + b ) = a 2 b 2 ( a b ) 2 = a 2 2 a b + b 2

We will use the special product formulas in the next few examples. We will start with the Binomial Squares formula.

Simplify: ( 2 + 3 ) 2 ( 4 2 5 ) 2 .

Solution

Be sure to include the 2 a b term when squaring a binomial.


  1. .
    Multiply using the binomial square pattern. .
    Simplify. .
    Combine like terms. .


  2. .
    Multiply using the binomial square pattern. .
    Simplify. .
    Combine like terms. .

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Simplify: ( 10 + 2 ) 2 ( 1 + 3 6 ) 2 .

102 + 20 2 55 + 6 6

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Simplify: ( 6 5 ) 2 ( 9 2 10 ) 2 .

31 12 5 121 36 10

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Simplify: ( 1 + 3 x ) 2 .

Solution

.
Multiply using the binomial square pattern. .
Simplify. .

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Simplify: ( 2 + 5 m ) 2 .

4 + 20 m + 25 m

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Simplify: ( 3 4 n ) 2 .

9 24 n + 16 n

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In the next two examples, we will find the product of conjugates.

Simplify: ( 4 2 ) ( 4 + 2 ) .

Solution

.
Multiply using the binomial square pattern. .
Simplify. .

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Simplify: ( 2 3 ) ( 2 + 3 ) .

1

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Simplify: ( 1 + 5 ) ( 1 5 ) .

−4

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Simplify: ( 5 2 3 ) ( 5 + 2 3 ) .

Solution

.
Multiply using the binomial square pattern. .
Simplify. .

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Simplify: ( 3 2 5 ) ( 3 + 2 5 ) .

−11

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Simplify: ( 4 + 5 7 ) ( 4 5 7 ) .

−159

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Access these online resources for additional instruction and practice with multiplying square roots.

Key concepts

  • Product Property of Square Roots If a , b are nonnegative real numbers, then
    a b = a · b and a · b = a b
  • Special formulas for multiplying binomials and conjugates:
    ( a + b ) 2 = a 2 + 2 a b + b 2 ( a b ) ( a + b ) = a 2 b 2 ( a b ) 2 = a 2 2 a b + b 2
  • The FOIL method can be used to multiply binomials containing radicals.

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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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