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Intermediate Algebra

8.1 Simplify Expressions with Roots

Intermediate Algebra8.1 Simplify Expressions with Roots
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  15. Index

Learning Objectives

By the end of this section, you will be able to:
  • Simplify expressions with roots
  • Estimate and approximate roots
  • Simplify variable expressions with roots
Be Prepared 8.1

Before you get started, take this readiness quiz.

  1. Simplify: (−9)2(−9)2 9292 (−9)3.(−9)3.
    If you missed this problem, review Example 2.21.
  2. Round 3.8463.846 to the nearest hundredth.
    If you missed this problem, review Example 1.34.
  3. Simplify: x3·x3x3·x3 y2·y2·y2y2·y2·y2 z3·z3·z3·z3.z3·z3·z3·z3.
    If you missed this problem, review Example 5.12.

Simplify Expressions with Roots

In Foundations, we briefly looked at square roots. Remember that when a real number n is multiplied by itself, we write n2n2 and read it ‘n squared’. This number is called the square of n, and n is called the square root. For example,

132is read “13 squared”169 is called thesquareof 13, since132=16913 is asquare rootof 169132is read “13 squared”169 is called thesquareof 13, since132=16913 is asquare rootof 169

Square and Square Root of a number

Square

Ifn2=m,thenmis thesquareofn.Ifn2=m,thenmis thesquareofn.

Square Root

Ifn2=m,thennis asquare rootofm.Ifn2=m,thennis asquare rootofm.

Notice (−13)2 = 169 also, so −13 is also a square root of 169. Therefore, both 13 and −13 are square roots of 169.

So, every positive number has two square roots—one positive and one negative. What if we only wanted the positive square root of a positive number? We use a radical sign, and write, m,m, which denotes the positive square root of m. The positive square root is also called the principal square root. This symbol, as well as other radicals to be introduced later, are grouping symbols.

We also use the radical sign for the square root of zero. Because 02=0,02=0, 0=0.0=0. Notice that zero has only one square root.

Square Root Notation

mis read “the square root ofm”.Ifn2=m,thenn=m,forn0.mis read “the square root ofm”.Ifn2=m,thenn=m,forn0.
The image shows the variable m inside a square root symbol. The symbol is a line that goes up along the left side and then flat above the variable. The symbol is labeled “radical sign”. The variable m is labeled “radicand”.

We know that every positive number has two square roots and the radical sign indicates the positive one. We write 169=13.169=13. If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, 169=−13.169=−13.

Example 8.1

Simplify: 144144 289.289.

Try It 8.1

Simplify: 6464 225.225.

Try It 8.2

Simplify: 100100 121.121.

Can we simplify −49?−49? Is there a number whose square is −49?−49?

()2=−49()2=−49

Any positive number squared is positive. Any negative number squared is positive. There is no real number equal to −49.−49. The square root of a negative number is not a real number.

Example 8.2

Simplify: −196−196 64.64.

Try It 8.3

Simplify: −169−169 81.81.

Try It 8.4

Simplify: 4949 −121.−121.

So far we have only talked about squares and square roots. Let’s now extend our work to include higher powers and higher roots.

Let’s review some vocabulary first.

We write:We say:n2nsquaredn3ncubedn4nto the fourth powern5nto the fifth powerWe write:We say:n2nsquaredn3ncubedn4nto the fourth powern5nto the fifth power

The terms ‘squared’ and ‘cubed’ come from the formulas for area of a square and volume of a cube.

It will be helpful to have a table of the powers of the integers from −5 to 5. See Figure 8.2.

The figure contains two tables. The first table has 9 rows and 5 columns. The first row is a header row with the headers “Number”, “Square”, “Cube”, “Fourth power”, and “Fifth power”. The second row contains the expressions n, n squared, n cubed, n to the fourth power, and n to the fifth power. The third row contains the number 1 in each column. The fourth row contains the numbers 2, 4, 8, 16, 32. The fifth row contains the numbers 3, 9, 27, 81, 243. The sixth row contains the numbers 4, 16, 64, 256, 1024. The seventh row contains the numbers 5, 25, 125 625, 3125. The eighth row contains the expressions x, x squared, x cubed, x to the fourth power, and x to the fifth power. The last row contains the expressions x squared, x to the fourth power, x to the sixth power, x to the eighth power, and x to the tenth power. The second table has 7 rows and 5 columns. The first row is a header row with the headers “Number”, “Square”, “Cube”, “Fourth power”, and “Fifth power”. The second row contains the expressions n, n squared, n cubed, n to the fourth power, and n to the fifth power. The third row contains the numbers negative 1, 1 negative 1, 1, negative 1. The fourth row contains the numbers negative 2, 4, negative 8, 16, negative 32. The fifth row contains the numbers negative 3, 9, negative 27, 81, negative 243. The sixth row contains the numbers negative 4, 16, negative 64, 256, negative 1024. The last row contains the numbers negative 5, 25, negative 125, 625, negative 3125.
Figure 8.2

Notice the signs in the table. All powers of positive numbers are positive, of course. But when we have a negative number, the even powers are positive and the odd powers are negative. We’ll copy the row with the powers of −2 to help you see this.

The image contains a table with 2 rows and 5 columns. The first row contains the expressions n, n squared, n cubed, n to the fourth power, and n to the fifth power. The second row contains the numbers negative 2, 4, negative 8, 16, negative 32. Arrows point to the second and fourth columns with the label “Even power Positive result”. Arrows point to the first third and fifth columns with the label “Odd power Negative result”.

We will now extend the square root definition to higher roots.

nth Root of a Number

Ifbn=a,thenbis annthroot ofa.The principalnthroot ofais writtenan.nis called theindexof the radical.Ifbn=a,thenbis annthroot ofa.The principalnthroot ofais writtenan.nis called theindexof the radical.

Just like we use the word ‘cubed’ for b3, we use the term ‘cube root’ for a3.a3.

We can refer to Figure 8.2 to help find higher roots.

43=6434=81(−2)5=−32643=4814=3−325=−243=6434=81(−2)5=−32643=4814=3−325=−2

Could we have an even root of a negative number? We know that the square root of a negative number is not a real number. The same is true for any even root. Even roots of negative numbers are not real numbers. Odd roots of negative numbers are real numbers.

Properties of anan

When n is an even number and

  • a0,a0, then anan is a real number.
  • a<0,a<0, then anan is not a real number.

When n is an odd number, anan is a real number for all values of a.

We will apply these properties in the next two examples.

Example 8.3

Simplify: 643643 814814 325.325.

Try It 8.5

Simplify: 273273 25642564 2435.2435.

Try It 8.6

Simplify: 1000310003 164164 2435.2435.

In this example be alert for the negative signs as well as even and odd powers.

Example 8.4

Simplify: −1253−1253 164164 −2435.−2435.

Try It 8.7

Simplify: −273−273 −2564−2564 −325.−325.

Try It 8.8

Simplify: −2163−2163 −814−814 −10245.−10245.

Estimate and Approximate Roots

When we see a number with a radical sign, we often don’t think about its numerical value. While we probably know that the 4=2,4=2, what is the value of 2121 or 503?503? In some situations a quick estimate is meaningful and in others it is convenient to have a decimal approximation.

To get a numerical estimate of a square root, we look for perfect square numbers closest to the radicand. To find an estimate of 11,11, we see 11 is between perfect square numbers 9 and 16, closer to 9. Its square root then will be between 3 and 4, but closer to 3.

The figure contains two tables. The first table has 5 rows and 2 columns. The first row is a header row with the headers “Number” and “Square Root”. The second row has the numbers 4 and 2. The third row is 9 and 3. The fourth row is 16 and 4. The last row is 25 and 5. A callout containing the number 11 is directed between the 9 and 16 in the first column. Another callout containing the number square root of 11 is directed between the 3 and 4 of the second column. Below the table are the inequalities 9 is less than 11 is less than 16 and 3 is less than square root of 11 is less than 4. The second table has 5 rows and 2 columns. The first row is a header row with the headers “Number” and “Cube Root”. The second row has the numbers 8 and 2. The third row is 27 and 3. The fourth row is 64 and 4. The last row is 125 and 5. A callout containing the number 91 is directed between the 64 and 125 in the first column. Another callout containing the number cube root of 91 is directed between the 4 and 5 of the second column. Below the table are the inequalities 64 is less than 91 is less than 125 and 4 is less than cube root of 91 is less than 5.

Similarly, to estimate 913,913, we see 91 is between perfect cube numbers 64 and 125. The cube root then will be between 4 and 5.

Example 8.5

Estimate each root between two consecutive whole numbers: 105105 433.433.

Try It 8.9

Estimate each root between two consecutive whole numbers:

3838 933933

Try It 8.10

Estimate each root between two consecutive whole numbers:

8484 15231523

There are mathematical methods to approximate square roots, but nowadays most people use a calculator to find square roots. To find a square root you will use the xx key on your calculator. To find a cube root, or any root with higher index, you will use the xyxy key.

When you use these keys, you get an approximate value. It is an approximation, accurate to the number of digits shown on your calculator’s display. The symbol for an approximation is and it is read ‘approximately’.

Suppose your calculator has a 10 digit display. You would see that

52.236067978rounded to two decimal places is52.249343.105422799rounded to two decimal places is9343.1152.236067978rounded to two decimal places is52.249343.105422799rounded to two decimal places is9343.11

How do we know these values are approximations and not the exact values? Look at what happens when we square them:

(2.236067978)2=5.000000002(2.24)2=5.0176(3.105422799)4=92.999999991(3.11)4=93.54951841(2.236067978)2=5.000000002(2.24)2=5.0176(3.105422799)4=92.999999991(3.11)4=93.54951841

Their squares are close to 5, but are not exactly equal to 5. The fourth powers are close to 93, but not equal to 93.

Example 8.6

Round to two decimal places: 1717 493493 514.514.

Try It 8.11

Round to two decimal places:

1111 713713 1274.1274.

Try It 8.12

Round to two decimal places:

1313 843843 984.984.

Simplify Variable Expressions with Roots

The odd root of a number can be either positive or negative. For example,

Three equivalent expressions are written: the cube root of 4 cubed, the cube root of 64, and 4. There are arrows pointing to the 4 that is cubed in the first expression and the 4 in the last expression labeling them as “same”. Three more equivalent expressions are also written: the cube root of the quantity negative 4 in parentheses cubed, the cube root of negative 64, and negative 4. The negative 4 in the first expression and the negative 4 in the last expression are labeled as being the “same”.

But what about an even root? We want the principal root, so 6254=5.6254=5.

But notice,

Three equivalent expressions are written: the fourth root of the quantity 5 to the fourth power in parentheses, the fourth root of 625, and 5. There are arrows pointing to the 5 in the first expression and the 5 in the last expression labeling them as “same”. Three more equivalent expressions are also written: the fourth root of the quantity negative 5 in parentheses to the fourth power in parentheses, the fourth root of 625, and 5. The negative 5 in the first expression and the 5 in the last expression are labeled as being the “different”.

How can we make sure the fourth root of −5 raised to the fourth power is 5? We can use the absolute value. |−5|=5.|−5|=5. So we say that when n is even ann=|a|.ann=|a|. This guarantees the principal root is positive.

Simplifying Odd and Even Roots

For any integer n2,n2,

when the indexnis oddann=awhen the indexnis evenann=|a|when the indexnis oddann=awhen the indexnis evenann=|a|

We must use the absolute value signs when we take an even root of an expression with a variable in the radical.

Example 8.7

Simplify: x2x2 n33n33 p44p44 y55.y55.

Try It 8.13

Simplify: b2b2 w33w33 m44m44 q55.q55.

Try It 8.14

Simplify: y2y2 p33p33 z44z44 q55.q55.

What about square roots of higher powers of variables? The Power Property of Exponents says (am)n=am·n.(am)n=am·n. So if we square am, the exponent will become 2m.

(am)2=a2m(am)2=a2m

Looking now at the square root,

a2mSince(am)2=a2m.(am)2Sincenis evenann=|a|.|am|Soa2m=|am|.a2mSince(am)2=a2m.(am)2Sincenis evenann=|a|.|am|Soa2m=|am|.

We apply this concept in the next example.

Example 8.8

Simplify: x6x6 y16.y16.

Try It 8.15

Simplify: y18y18 z12.z12.

Try It 8.16

Simplify: m4m4 b10.b10.

The next example uses the same idea for highter roots.

Example 8.9

Simplify: y183y183 z84.z84.

Try It 8.17

Simplify: u124u124 v153.v153.

Try It 8.18

Simplify: c205c205 d246d246

In the next example, we now have a coefficient in front of the variable. The concept a2m=|am|a2m=|am| works in much the same way.

16r22=4|r11|because(4r11)2=16r22.16r22=4|r11|because(4r11)2=16r22.

But notice 25u8=5u425u8=5u4 and no absolute value sign is needed as u4 is always positive.

Example 8.10

Simplify: 16n216n2 81c2.81c2.

Try It 8.19

Simplify: 64x264x2 100p2.100p2.

Try It 8.20

Simplify: 169y2169y2 121y2.121y2.

This example just takes the idea farther as it has roots of higher index.

Example 8.11

Simplify: 64p6364p63 16q124.16q124.

Try It 8.21

Simplify: 27x27327x273 81q284.81q284.

Try It 8.22

Simplify: 125q93125q93 243q255.243q255.

The next examples have two variables.

Example 8.12

Simplify: 36x2y236x2y2 121a6b8121a6b8 64p63q93.64p63q93.

Try It 8.23

Simplify: 100a2b2100a2b2 144p12q20144p12q20 8x30y1238x30y123

Try It 8.24

Simplify: 225m2n2225m2n2 169x10y14169x10y14 27w36z15327w36z153

Media Access Additional Online Resources

Access this online resource for additional instruction and practice with simplifying expressions with roots.

Section 8.1 Exercises

Practice Makes Perfect

Simplify Expressions with Roots

In the following exercises, simplify.

1.

6464 8181

2.

169169 100100

3.

196196 11

4.

144144 121121

5.

4949 0.010.01

6.

6412164121 0.160.16

7.

−121−121 289289

8.

400400 −36−36

9.

225225 −9−9

10.

−49−49 256256

11.

21632163 25642564

12.

273273 164164 24352435

13.

51235123 814814 1515

14.

12531253 1296412964 1024510245

15.

−83−83 −814−814 −325−325

16.

−643−643 −164−164 −2435−2435

17.

−1253−1253 −12964−12964 −10245−10245

18.

−5123−5123 −814−814 −15−15

Estimate and Approximate Roots

In the following exercises, estimate each root between two consecutive whole numbers.

19.

7070 713713

20.

5555 11931193

21.

200200 13731373

22.

172172 20032003

In the following exercises, approximate each root and round to two decimal places.

23.

1919 893893 974974

24.

2121 933933 10141014

25.

5353 14731473 45244524

26.

4747 16331633 52745274

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

27.

u55u55 v88v88

28.

a33a33 b99b99

29.

y44y44 m77m77

30.

k88k88 p66p66

31.

x6x6 y16y16

32.

a14a14 w24w24

33.

x24x24 y22y22

34.

a12a12 b26b26

35.

x93x93 y124y124

36.

a105a105 b273b273

37.

m84m84 n205n205

38.

r126r126 s303s303

39.

49x249x2 81x1881x18

40.

100y2100y2 100m32100m32

41.

121m20121m20 64a264a2

42.

81x3681x36 25x225x2

43.

16x8416x84 64y12664y126

44.

−8c93−8c93 125d153125d153

45.

216a63216a63 32b20532b205

46.

128r147128r147 81s24481s244

47.

144x2y2144x2y2 169w8y10169w8y10 8a51b638a51b63

48.

196a2b2196a2b2 81p24q681p24q6 27p45q9327p45q93

49.

121a2b2121a2b2 9c8d129c8d12 64x15y66364x15y663

50.

225x2y2z2225x2y2z2 36r6s2036r6s20 125y18z273125y18z273

Writing Exercises

51.

Why is there no real number equal to −64?−64?

52.

What is the difference between 9292 and 9?9?

53.

Explain what is meant by the nth root of a number.

54.

Explain the difference of finding the nth root of a number when the index is even compared to when the index is odd.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “simplify expressions with roots.”, “estimate and approximate roots”, and “simplify variable expressions with roots”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

If most of your checks were:

…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.

…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?

…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.

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