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To simplify a square root expression that does not involve a fraction, we can use the following two rules:

Simplifying square roots without fractions

  1. If a factor of the radicand contains a variable with an even exponent, the square root is obtained by dividing the exponent by 2.
  2. If a factor of the radicand contains a variable with an odd exponent, the square root is obtained by first factoring the variable factor into two factors so that one has an even exponent and the other has an exponent of 1, then using the product property of square roots.

Sample set a

Simplify each square root.

a 4 .     The exponent is even: 4 2 = 2. The exponent on the square root is 2.

a 4 = a 2

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a 6 b 10 .    Both exponents are even: 6 2 = 3 and 10 2 = 5. The exponent on the square root of a 6 is 3. The exponent on the square root if b 10 is 5.

a 6 b 10 = a 3 b 5

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y 5 .    The exponent is odd: y 5 = y 4 y . Then

y 5 = y 4 y = y 4 y = y 2 y

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36 a 7 b 11 c 20 = 6 2 a 6 a b 10 b c 20 a 7 = a 6 a , b 11 = b 10 b = 6 2 a 6 b 10 c 20 · a b by the commutative property of multiplication. = 6 2 a 6 b 10 c 20 a b by the product property of square roots. = 6 a 3 b 5 c 10 a b

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49 x 8 y 3 ( a 1 ) 6 = 7 2 x 8 y 2 y ( a 1 ) 6 = 7 2 x 8 y 2 ( a 1 ) 6 y = 7 x 4 y ( a 1 ) 3 y

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75 = 25 · 3 = 5 2 · 3 = 5 2 3 = 5 3

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Practice set a

Simplify each square root.

81 a 12 b 6 c 38

9 a 6 b 3 c 19

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144 x 4 y 80 ( b + 5 ) 16

12 x 2 y 40 ( b + 5 ) 8

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w 7 z 3 k 13

w 3 z k 6 w z k

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27 a 3 b 4 c 5 d 6

3 a b 2 c 2 d 3 3 a c

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180 m 4 n 15 ( a 12 ) 15

6 m 2 n 7 ( a 12 ) 7 5 n ( a 12 )

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Square roots involving fractions

A square root expression is in simplified form if there are

  1. no perfect squares in the radicand,
  2. no fractions in the radicand, or
  3. 3. no square root expressions in the denominator.

The square root expressions 5 a , 4 3 x y 5 , and 11 m 2 n a 4 2 x 2 are in simplified form.

The square root expressions 3 x 8 , 4 a 4 b 3 5 , and 2 y 3 x are not in simplified form.

Simplifying square roots with fractions

To simplify the square root expression x y ,

  1. Write the expression as x y using the rule x y = x y .
  2. Multiply the fraction by 1 in the form of y y .
  3. Simplify the remaining fraction, x y y .

Rationalizing the denominator

The process involved in step 2 is called rationalizing the denominator. This process removes square root expressions from the denominator using the fact that ( y ) ( y ) = y .

Sample set b

Simplify each square root.

3 5 = 3 5 = 3 5 · 5 5 = 15 5

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9 8 = 9 8 = 9 8 · 8 8 = 3 8 8 = 3 4 · 2 8 = 3 4 2 8 = 3 · 2 2 8 = 3 2 4

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k 2 m 3 = k 2 m 3 = k m 3 = k m 2 m = k m 2 m = k m m = k m m · m m = k m m m m = k m m · m = k m m 2

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x 2 8 x + 16 = ( x 4 ) 2 = x 4

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Practice set b

Simplify each square root.

32 a 5 b 7

4 a 2 2 a b b 4

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Exercises

For the following problems, simplify each of the radical expressions.

100 x 5 y 11

10 x 2 y 5 x y

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5 16 m 6 n 7

20 m 3 n 3 n

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50 a 3 b 9

5 a b 4 2 a b

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4 18 a 5 b 17

12 a 2 b 8 2 a b

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4 75 a 4 b 6

20 a 2 b 3 3

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9 a 2 b 3

3 a b b

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m 6 n 8 p 12 q 20

m 3 n 4 p 6 q 10

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49 x 2 y 5 z 9 25 a 3 b 11

7 x y 2 z 4 a b y z 5 a 2 b 6

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( a 3 ) 4 ( a 1 ) 2

( a 3 ) 2 ( a 1 )

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( b + 7 ) 8 ( b 7 ) 6

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a 2 10 a + 25

( a 5 )

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( a 2 2 a + 1 ) 4

( a 1 ) 4

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Exercises for review

( [link] ) Solve the inequality 3 ( a + 2 ) 2 ( 3 a + 4 )

a 2 3

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( [link] ) Graph the inequality 6 x 5 ( x + 1 ) 6.
A horizontal line with arrows on both ends.

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( [link] ) Supply the missing words. When looking at a graph from left-to-right, lines with _______ slope rise, while lines with __________ slope fall.

positive; negative

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( [link] ) Simplify the complex fraction 5 + 1 x 5 1 x .

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( [link] ) Simplify 121 x 4 w 6 z 8 by removing the radical sign.

11 x 2 w 3 z 4

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Questions & Answers

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The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
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1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
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Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
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Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
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x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
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Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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