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Mathematics LibreTexts

9.3: Simplifying Square Root Expressions

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To begin our study of the process of simplifying a square root expression, we must note three facts: one fact concerning perfect squares and two concerning properties of square roots.

Perfect Squares

Rea numbers that are squares of rational numbers are called perfect squares. The numbers 25 and 14 are examples of perfect squares since 25=52 and 14=(12)2, and 5 and 12 are rational numbers. The number 2 is not a perfect square since 2=(2)2 and 2 is not a rational number.

Although we will not make a detailed study of irrational numbers, we will make the following observation:

Note

Any indicated square root whose radicand is not a perfect square number is an irrational number.

The numbers 6,15 and 34 are each irrational since each radicand 6,15,34 is not a perfect square.

The Product Property of Square Roots

Notice that

94=36=6 and 94=32=6

The Product Property xy=xy

This suggests that in general, if x and y are positive real numbers,

xy=xy

The square root of the product is the product of the square roots.

The Quotient Property of Square Roots

We can suggest a similar rule for quotients. Notice that

364=9=3 and

364=62=3.

Since both 364 and 364 equal 3, it must be that

364=364

The Quotient Property xy=xy

This suggests that in general, if x and y are positive real numbers,

xy=xy,y0.

The square root of the quotient is the quotient of the square roots.

CAUTION

It is extremely important to remeber that

x+yx+y or xyxy

For example, notice that 16+9=25=5, but 16+9=4+3=7

We shall study the process of simplifying a square root expresion by distinguishing between two types of square roots: square roots not involving a fraction and square roots involving a fraction.

Square Roots Not Involving Fractions

A square root that does not involve fractions is in the simplified form if there is no perfect square in the radicand.

The square roots x,.ab,5mn,2(a+5) are in simplified form since none of the radicands contains a perfect square.

The square roots x2,a3=a2a are not in simplified form since each radicand contains a perfect square.

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.

Example 9.3.1

a4. The exponent is even: 42=2. The exponent on the square root is 2.

a4=a2

Example 9.3.2

a6b10. Both exponents are even: 62=3 and 102=5. The exponent on the square root of a6 is 3. The exponent on the square root if b10 is 5.

a6gb10=a3b5

Example 9.3.3

y5. The exponent is odd: y5=y4y. The

y5=y4y=y4y=y2y

Example 9.3.4

36a7b11c20=62a6ab10bc20a7=a6a,b11=b10b=62a6b10c20ab by the commutative property of multiplication =62a6b10c20ab by the product property of square roots =6a3b5c10ab

Example 9.3.5

49x8y3(a1)6=72x8y2y(a1)6=72x8y2(a1)6y=7x4y(a1)3y

Example 9.3.6

75=253=523=523=53

Practice Set A

Simplify each square root.

Practice Problem 9.3.1

m8

Answer

m4

Practice Problem 9.3.2

h14k22

Answer

h7k11

Practice Problem 9.3.3

81a12b6c38

Answer

9a6b3c19

Practice Problem 9.3.4

144x4y80(b+5)16

Answer

12x2y40(b+5)8

Practice Problem 9.3.5

w5

Answer

w2w

Practice Problem 9.3.6

w7z3k13

Answer

w3zk6wzk

Practice Problem 9.3.7

27a3b4c5d6

Answer

3ab2c2d33ac

Practice Problem 9.3.8

180m4n159a12)15

Answer

6m2n7(a12)75n(a12)

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. no square root expressions in the denominator.

The square root expressions 5a,43xy5, and 11m2na42x2 are in simplified form

The square root expressions 3x8,4a4b35, and 2y3x are not in simplified form.

Simplifying Square Roots with Fractions

To simplify the square root expression xy,

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

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.

Example 9.3.7

925=925=35

Example 9.3.8

35=35=3555=155

Example 9.3.9

98=98=9888=388=3428=3428=3228=324

Example 9.3.10

k2m3=k2m3=km3=km2m=km2m=kmm=kmmmm=kmmmm=kmmm=kmm2

Example 9.3.11

x28x+16=(x4)2=x4

Practice Set B

Simplify each square root.

Practice Problem 9.3.9

8125

Answer

95

Practice Problem 9.3.10

27

Answer

147

Practice Problem 9.3.11

45

Answer

255

Practice Problem 9.3.12

104

Answer

102

Practice Problem 9.3.13

94

Answer

32

Practice Problem 9.3.14

a36

Answer

a6a6

Practice Problem 9.3.15

y4x3

Answer

y2xx2

Practice Problem 9.3.16

32a5b7

Answer

4a22abb4

Practice Problem 9.3.17

(x+9)2

Answer

x+9

Practice Problem 9.3.18

x2+14x+49

Answer

x+7

Exercises

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

Exercise 9.3.1

8b2

Answer

2b2

Exercise 9.3.2

20a2

Exercise 9.3.3

24x4

Answer

2x26

Exercise 9.3.4

27y6

Exercise 9.3.5

a5

Answer

a2a

Exercise 9.3.6

m7

Exercise 9.3.7

x11

Answer

x5x

Exercise 9.3.8

y17

Exercise 9.3.9

36n9

Answer

6n4n

Exercise 9.3.10

49x13

Exercise 9.3.11

100x5y11

Answer

10x2y5xy

Exercise 9.3.12

64a7b3

Exercise 9.3.13

516m6n7

Answer

20m3n3n

Exercise 9.3.14

89a4b11

Exercise 9.3.15

316x3

Answer

12xx

Exercise 9.3.16

825y3

Exercise 9.3.17

12a4

Answer

2a23

Exercise 9.3.18

32x7

Answer

4x32x

Exercise 9.3.19

12y13

Exercise 9.3.20

50a3b9

Answer

5ab42ab

Exercise 9.3.21

48p11q5

Exercise 9.3.22

418a5b17

Answer

12a2b82ab

Exercise 9.3.23

8108x21y3

Exercise 9.3.24

475a4b6

Answer

20a2b33

Exercise 9.3.25

672x2y4z10

Exercise 9.3.26

b12

Answer

b6

Exercise 9.3.27

c18

Exercise 9.3.28

a2b2c2

Answer

abc

Exercise 9.3.29

4x2y2z2

Exercise 9.3.30

9a2b3

Answer

3abb

Exercise 9.3.31

16x4y5

Exercise 9.3.32

m6n8p12q20

Answer

m3n4p6q10

Exercise 9.3.33

r2

Exercise 9.3.34

p2

Answer

p

Exercise 9.3.35

14

Exercise 9.3.36

116

Answer

14

Exercise 9.3.37

425

Exercise 9.3.38

949

Answer

37

Exercise 9.3.39

583

Exercise 9.3.40

2323

Answer

863

Exercise 9.3.41

56

Exercise 9.3.42

27

Answer

147

Exercise 9.3.43

310

Exercise 9.3.44

43

Answer

233

Exercise 9.3.45

25

Exercise 9.3.46

310

Answer

3010

Exercise 9.3.47

16a25

Exercise 9.3.48

24a57

Answer

2a242a7

Exercise 9.3.49

72x2y35

Exercise 9.3.50

2a

Answer

2aa

Exercise 9.3.51

5b

Exercise 9.3.52

6x3

Answer

6xx2

Exercise 9.3.53

12y5

Exercise 9.3.54

49x2y5z925a3b11

Answer

7xy2z4abyz5a2b6

Exercise 9.3.55

27x6y1533x3y5

Exercise 9.3.56

(b+2)4

Answer

(b+2)2

Exercise 9.3.57

(a7)8

Exercise 9.3.58

(x+2)6

Answer

(x+2)3

Exercise 9.3.59

(x+2)2(x+1)2

Exercise 9.3.60

(a3)4(a1)2

Answer

(a3)2(a1)

Exercise 9.3.61

(b+7)8(b7)6

Exercise 9.3.62

a210a+25

Answer

(a5)

Exercise 9.3.63

b2+6b+9

Exercise 9.3.64

(a22a+1)4

Answer

(a1)4

Exercise 9.3.65

(x2+2x+1)12

Exercises For Review

Exercise 9.3.66

Solve the inequality 3(a+2)2(3a+4)

Answer

a23

Exercise 9.3.67

Graph the inequality 6x5(x+1)6

A horizontal line with arrows on both ends.

Exercise 9.3.68

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

Answer

positive; negative

Exercise 9.3.69

Simplify the complex fraction: 5+1x51x

Exercise 9.3.70

Simplify 121x4w6z8 by removing the radical sign.

Answer

11x2w3z4


This page titled 9.3: Simplifying Square Root Expressions is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Denny Burzynski & Wade Ellis, Jr. (OpenStax CNX) .

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