Mathematics
Simplifying Radicals
Simplifying radicals is the process of finding the simplest form of a radical expression. This involves factoring out perfect squares from the radicand and simplifying any remaining factors. The simplified form of a radical expression has no radicals in the denominator and no perfect square factors under the radical sign.
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11 Key excerpts on "Simplifying Radicals"
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
A radical expression is in simplified form if 1. None of the factors of the radicand (the quantity under the radical sign) can be written as powers greater than or equal to the index — that is, no perfect squares can be factors of the quantity under a square root sign, no perfect cubes can be factors of what is under a cube root sign, and so forth. 2. There are no fractions under the radical sign. 3. There are no radicals in the denominator. RULE Simplified Form for Radical Expressions EXAMPLE 1 VIDEO EXAMPLES SECTION 7.3 EXAMPLE 2 Note In Example 2, we do not want to write 24 as 4 ∙ 6, because the problem involves a cube root, not a square root. Neither 4 nor 6 are perfect cubes. 7.3 Simplified Form for Radicals 511 Write in simplified form: √ — 48 x 4 y 3 , where x , y ≥ 0 SOLUTION The largest perfect square that is a factor of the radicand is 16 x 4 y 2 . Applying Property 2, we have √ — 48 x 4 y 3 = √ — 16 x 4 y 2 ⋅ 3 y = √ — 16 x 4 y 2 √ — 3 y = 4 x 2 y √ — 3 y Write 3 √ — 40 a 5 b 4 in simplified form. SOLUTION We now want to factor the largest perfect cube from the radicand. We write 40 a 5 b 4 as 8 a 3 b 3 ⋅ 5 a 2 b and proceed as in previous examples. 3 √ — 40 a 5 b 4 = 3 √ — 8 a 3 b 3 ⋅ 5 a 2 b = 3 √ — 8 a 3 b 3 3 √ — 5 a 2 b = 2 ab 3 √ — 5 a 2 b Fractions and Radical Expressions We now consider some examples that involve fractions and simplified form for radicals. Simplify each expression. a. √ — 12 ____ 6 b. 5 √ — 18 _____ 15 c. 6 + √ — 8 _______ 2 d. − 1 + √ — 45 _________ 2 SOLUTION These expressions are not yet simplified because they do not meet the first condition for simplified form. In each case, we simplify the radical first, then we factor and reduce to lowest terms. a. √ — 12 ____ 6 = 2 √ — 3 ____ 6 Simplify the radical √ — 12 = √ — 4∙3 = √ — 4 √ — 3 = 2 √ — 3 = 2 √ — 3 ____ 2 ⋅ 3 Factor denominator = √ — 3 ____ 3 Divide out common factors b.
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2017(Publication Date)
- XYZ Textbooks(Publisher)
To do this, we use the first of two properties of radicals. Consider the following two problems: √ — 9 ⋅ 16 = √ — 144 = 12 √ — 9 ⋅ √ — 16 = 3 ⋅ 4 = 12 Since the answers to both are equal, the original problems also must be equal; that is, √ — 9 ⋅ 16 = √ — 9 ⋅ √ — 16 . We can generalize this property as follows. A radical expression is in simplified form if 1. There are no perfect squares that are factors of the quantity under the square root sign, no perfect cubes that are factors of the quantity under the cube root sign, and so on. We want as little as possible under the radical sign. 2. There are no fractions under the radical sign. 3. There are no radicals in the denominator. simplified form DEFINITION Note Simplified form for radicals is the form that we work toward when Simplifying Radicals. The properties of radicals are the tools we use to get us to simplified form. Simplified Form and Properties of Radicals 536 Chapter 8 Roots and Radical Expressions We can use this property to simplify a radical expression whose radicand contains a perfect square. Simplify: √ — 20 . SOLUTION To simplify √ — 20, we want to take as much out from under the radical sign as possible. We begin by looking for the largest perfect square that is a factor of 20. The largest perfect square that divides 20 is 4, so we write 20 as 4 ⋅ 5: √ — 20 = √ — 4 ⋅ 5 Next, we apply the product property of radicals and write √ — 4 ⋅ 5 = √ — 4 √ — 5 And since √ — 4 = 2, we have √ — 4 √ — 5 = 2 √ — 5 The expression 2 √ — 5 is the simplified form of √ — 20 since we have taken as much out from under the radical sign as possible. Simplify: √ — 75 . SOLUTION Since 25 is the largest perfect square that divides 75, we have √ — 75 = √ — 25 ⋅ 3 Factor 75 into 25 ⋅ 3 = √ — 25 √ — 3 Product property xfor square roots = 5 √ — 3 √ — 25 = 5 The expression 5 √ — 3 is the simplified form for √ — 75 since we have taken as much out from under the radical sign as possible.
eBook - PDFIntroductory Algebra
Concepts and Graphs 2E
- Charles P. McKeague(Author)
- 2020(Publication Date)
- XYZ Textbooks(Publisher)
476 CHAPTER 8 Roots and Radicals Getting Ready for the Next Section Simplify. 73. √ — 4x 3 y 2 74. √ — 9x 2 y 3 75. 6 __ 2 √ — 16 76. 8 __ 4 √ — 9 77. √ — 2 _ √ — 4 78. √ — 6 _ √ — 9 79. 3 √ — 18 _ 3 √ — 9 80. 3 √ — 12 _ 3 √ — 8 Multiply. 81. √ — 2 _ √ — 3 ⋅ √ — 3 _ √ — 3 82. √ — y _ √ — 2 ⋅ √ — 2 _ √ — 2 83. 3 √ — 3 ⋅ 3 √ — 9 84. 3 √ — 4 ⋅ 3 √ — 2 477 Simplified Form of Radicals Radical expressions that are in simplified form are generally the easiest form to work with. A radical expression is in simplified form if it has three special characteristics. A radical expression is in simplified form if 1. There are no perfect squares that are factors of the quantity under the square root sign, no perfect cubes that are factors of the quantity under the cube root sign, and so on. We want as little as possible under the radical sign. 2. There are no fractions under the radical sign. 3. There are no radicals in the denominator. DEFINITION: SIMPLIFIED FORM A radical expression that has these three characteristics is said to be in sim- plified form. As we will see, simplified form is not always the least complicated expression. In many cases, the simplified expression looks more complicated than the original expression. The important thing about simplified form for radicals is that simplified expressions are easier to work with. A Properties of Radicals The tools we will use to put radical expressions into simplified form are the properties of radicals. We list the properties again for clarity. If a and b represent any two nonnegative real numbers, then it is always true that 1. √ — a √ — b = √ — a ⋅ b 2. √ — a _ √ — b = √ __ a __ b b ≠ 0 3. √ — a √ — a = ( √ — a ) 2 = a This property comes directly from the definition of radicals PROPERTY: PROPERTIES OF RADICALS The following examples illustrate how we put a radical expression into simplified form using the three properties of radicals. Although the properties are stated for square roots only, they hold for all roots.
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
Here are some examples: 3 √ — 2 1 __ 2 √ — 5 5 3 √ — 7 3 x √ — 2 x 2 a 2 b 3 √ — 5 a All of these are products. The first expression 3 √ — 2 is the product of 3 and √ — 2. That is, 3 √ — 2 = 3 ⋅ √ — 2 The 3 and the √ — 2 are not stuck together in some mysterious way. The expression 3 √ — 2 is simply the product of two numbers, one of which is rational, and the other is irrational. A radical expression is in simplified form if 1. None of the factors of the radicand (the quantity under the radical sign) can be written as powers greater than or equal to the index — that is, no perfect squares can be factors of the quantity under a square root sign, no perfect cubes can be factors of what is under a cube root sign, and so forth. 2. There are no fractions under the radical sign. 3. There are no radicals in the denominator. RULE Simplified Form for Radical Expressions EXAMPLE 1 VIDEO EXAMPLES SECTION 10.3 EXAMPLE 2 Note In Example 2, we do not want to write 24 as 4 ∙ 6, because the problem involves a cube root, not a square root. Neither 4 nor 6 are perfect cubes. 10.3 Simplified Form for Radicals 721 Write in simplified form: √ — 48 x 4 y 3 , where x , y ≥ 0 SOLUTION The largest perfect square that is a factor of the radicand is 16 x 4 y 2 . Applying Property 2, we have √ — 48 x 4 y 3 = √ — 16 x 4 y 2 ⋅ 3 y = √ — 16 x 4 y 2 √ — 3 y = 4 x 2 y √ — 3 y Write 3 √ — 40 a 5 b 4 in simplified form. SOLUTION We now want to factor the largest perfect cube from the radicand. We write 40 a 5 b 4 as 8 a 3 b 3 ⋅ 5 a 2 b and proceed as in previous examples. 3 √ — 40 a 5 b 4 = 3 √ — 8 a 3 b 3 ⋅ 5 a 2 b = 3 √ — 8 a 3 b 3 3 √ — 5 a 2 b = 2 ab 3 √ — 5 a 2 b Fractions and Radical Expressions We now consider some examples that involve fractions and simplified form for radicals. Simplify each expression. a. √ — 12 ____ 6 b. 5 √ — 18 _____ 15 c.
eBook - PDFBeginning Algebra
Connecting Concepts through Applications
- Mark Clark, Cynthia Anfinson(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
There are also excluded values for radical expressions, namely, those values that result in negative numbers under the square root. Copyright 2019 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. C H A P T E R 8 R a d i c a l E x p r e s s i o n s a n d E q u a t i o n s 684 Evaluating Radical Equations Square roots often show up in applications and equations. A radical equation is an equation in which the variable is in the radicand (under the radical). DEFINITION Radical Equation A radical equation is an equation in which a variable expression is under the radical. For instance, ! x 2 6 5 2 is a radical equation. We now learn how to evaluate radical equations. Solving radical equations is covered in Section 8.4. Example 5 Using a formula to model the height of a falling object A science class tests how long it takes a ball to fall to the ground when dropped from various heights. The class determines that the time, T , in seconds that it takes the ball to fall to the ground after being dropped from a height of h feet is given by the equation T 5 0.243 ! h a. What units does h represent? What units does T represent? b. How long will it take the ball to fall to the ground when the ball is dropped from 15 feet? Round the answer to the hundredths place if necessary. c. How long will it take the ball to fall to the ground when the ball is dropped from 50 feet? Round the answer to the hundredths place if necessary. SOLUTION a. T , in seconds, is the time it takes the ball to fall to the ground. The variable h , in feet, is the initial height the ball is dropped from.
eBook - PDF- Alan Tussy, R. Gustafson(Authors)
- 2012(Publication Date)
- Cengage Learning EMEA(Publisher)
Except for 1, the radicand has no perfect-square factors. 2. No fraction appears in the radicand. 3. No radical appears in the denominator. These radicals are not in simplified form: The radicand has a perfect-The radicand is A radical appears square factor of . a fraction. in the denominator. 9 m 2 1 2 3 B 5 64 2 18 m 3 The product and quotient rules for square roots can be used to simplify radical expressions. B a b 1 a 2 b ( b 0) 2 a b 1 a 2 b Simplify: Write as a product of its greatest perfect-square factor, , and one other factor. The square root of a product is equal to the product of the square roots. Find the square root of the perfect square: . Simplify: The square root of a quotient is equal to the quotient of the square roots. In the denominator, evaluate: . 2 64 8 2 5 8 B 5 64 2 5 2 64 2 9 m 2 3 m 3 m 2 2 m 2 9 m 2 2 2 m 9 m 2 18 m 3 2 18 m 3 2 9 m 2 2 m Simplify. All variables represent positive real numbers. 23. 24. 25. 26. 27. 28. 29. 30. 31. Fitness Equipment. The length of the sit-up board can be found using the Pythagorean theorem. Find its length. Express the answer in simplified radical form. Then express your result as a decimal approximation rounded to the nearest tenth. B 242 x 4 169 x 2 B 60 49 B 16 25 2 250 t 3 2 2 63 2 80 x 2 2 x 5 2 32 32. Determine whether the statement is true or false: . 2 x 9 x 3 6 ft 2 ft REVIEW EXERCISES 12. Road Signs. To find the maximum velocity a car can safely travel around a curve without skidding, we can use the formula , where is the velocity in mph and is the radius of the curve in feet. How should the road sign be labeled if it is to be posted in front of a curve with a radius of 360 feet? Simplify. All variables represent positive real numbers. 13. 14. 15. 16. Refer to the right triangle. 17. Find where and . 18. Find where and . c 17 a 8 b b 8 a 6 c 2 9 h 16 2 y 12 2 4 b 4 2 x 2 ? mph r v v 2 2.5 r 19. Theater Seating. How much higher is the seat at the top of the incline than the one at the bottom? 20.
eBook - PDFBeginning and Intermediate Algebra
A Guided Approach
- Rosemary Karr, Marilyn Massey, R. Gustafson, , Rosemary Karr, Marilyn Massey, R. Gustafson(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 648 CHAPTER 9 Radicals and Rational Exponents To express square roots, we use the symbol , called a radical sign . For example, 121 5 11 Read as “The positive square root of 121 is 11.” 2 121 5 2 11 Read as “The negative square root of 121 is 2 11 .” The number under the radical sign is called a radicand . Comment The principal square root of a positive number is always positive. Although 5 and 2 5 are both square roots of 25, only 5 is the principal square root. The radical expression 25 represents 5. The radical expression 2 25 represents 2 5 . Teaching Tip Christoff Rudolff (1500–1545?) was the first to use the ! symbol. SQUARE ROOT OF a If a . 0 , ! a is the positive number, the principal square root of a , whose square is a . In symbols, 1! a 2 2 5 a If a 5 0 , ! a 5 0 5 0 . The principal square root of 0 is 0. If a , 0 , ! a is not a real number. Because of the previous definition, the square root of any number squared is that number. For example, 1 10 2 2 5 10 ? 10 5 10 1! a 2 2 5 ! a ? ! a 5 a Simplify each radical. a. 1 5 1 b. 81 5 9 c. 2 81 5 2 9 d. 2 225 5 2 15 e. Å 1 4 5 1 2 f. 2 Å 16 121 5 2 4 11 g. 0.04 5 0.2 h. 2 0.0009 5 2 0.03 Simplify: a. 2 49 2 7 b. Å 25 49 5 7 c. 0.0036 0.06 Numbers such as 1, 4, 9, 16, 49 , and 1,600 are called integer squares , because each one is the square of an integer. The square root of every integer square is an integer. 1 5 1 4 5 2 9 5 3 16 5 4 49 5 7 1,600 5 40 EXAMPLE 2 a SELF CHECK 2 Perspective CALCULATING SQUARE ROOTS The Bakhshali manuscript is an early mathematical manu-script that was discovered in India in the late 19th century.
eBook - PDFIntroductory Algebra
Concepts with Applications
- Charles P. McKeague(Author)
- 2013(Publication Date)
- XYZ Textbooks(Publisher)
Circle the mistake and write the correct word(s) or expression on the line provided. 1. When finding the product of 5 √ — 6 and 7 √ — 2 , we can set (5 √ — 6 )(7 √ — 2 ) equal to ( √ — 5 ⋅ 7)( √ — 6 ⋅ 2 ). 2. Multiplying conjugates of the form √ — a + √ — b and √ — a − √ — b will always produce an irrational number. 3. The conjugate of the expression 2 √ — 11 + √ — 6 is 2 √ — 66. 4. To rationalize the denominator of the expression √ — 7 __ ( √ — 5 − √ — 3) , you must multiply the numerator and the denominator by √ — 7. OBJECTIVES 606 KEY WORDS Chapter 8 Radical Expressions and Equations squaring property of equality extraneous solutions radical equation A Solve equations containing radicals. B Graph simple square root and cube root equations in two variables. 8.5 Radical Equations and Graphs Imagine you are standing 220 feet above San Francisco Bay on the Golden Gate Bridge. If you accidentally drop your car keys over the rail, how long will it take them to reach 10 feet above the water? How long before they hit the surface of the water? If we assign the surface of the water a value of 0, both answers can be found by solving the following equation: t = √ — 220 − h _________ 4 where t is the time of the keys’ fall and h is their height at any given time. Solving radical equations like this is our focus in this section. We will work with equations that involve one or more radicals. The first step in solving an equation that contains a radical is to eliminate the radical from the equation. To do so, we need an additional property. A Solving Equations with Radicals We will never lose solutions to our equations by squaring both sides. We may, however, introduce extraneous solutions. Extraneous solutions satisfy the equation obtained by squaring both sides of the original equation, but do not satisfy the original equation. We know that if two real numbers a and b are equal, then so are their squares.
eBook - PDFIntermediate Algebra
A Guided Approach
- Rosemary Karr, Marilyn Massey, R. Gustafson, , Rosemary Karr, Marilyn Massey, R. Gustafson(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 502 CHAPTER 7 Radical Expressions and Equations In this section, we will solve equations that contain radicals. To do so, we will use the power rule . Then, we will use this rule to solve an application. Solve a radical equation containing one radical. 1 Simplify. 1. 1 a 2 2 2. 1 5 x 2 2 3. 1 x 1 4 2 2 4. 1 3 y 2 3 2 3 THE POWER RULE If x , y , and n are real numbers and x 5 y , then x n 5 y n When we raise both sides of an equation to the same power, the resulting equation might not be equivalent to the original equation. For example, if we square both sides of the equation x 5 3 With a solution set of 5 3 6 we obtain the equation x 2 5 9 With a solution set of 5 3, 2 3 6 Equations 1 and 2 are not equivalent, because they have different solution sets, and the solution 2 3 of Equation 2 does not satisfy Equation 1. Since raising both sides of an equa-tion to the same power can produce an equation with roots (extraneous) that do not satisfy the original equation, we must check each possible solution in the original equation. Solve: x 1 3 5 4 Because the radical term is isolated, to eliminate the radical, we apply the power rule by squaring both sides of the equation, and proceed as follows: x 1 3 5 4 1 x 1 3 2 2 5 1 4 2 2 Square both sides of the equation. x 1 3 5 16 Simplify. x 5 13 Subtract 3 from both sides. To check the apparent solution of 13, we can substitute 13 for x and determine whether it satisfies the original equation. (1) (2) EXAMPLE 1 Solution power rule Vocabulary Getting Ready Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience.
eBook - PDF- Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis(Authors)
- 2020(Publication Date)
- Openstax(Publisher)
Trying to add an integer and a radical is like trying to add an integer and a variable—they are not like terms! TRY IT : : 9.35 Simplify: 5 + 75 . TRY IT : : 9.36 Simplify: 2 + 98 . The next example includes a fraction with a radical in the numerator. Remember that in order to simplify a fraction you need a common factor in the numerator and denominator. 1046 Chapter 9 Roots and Radicals This OpenStax book is available for free at http://cnx.org/content/col31130/1.4 EXAMPLE 9.19 Simplify: 4 − 48 2 . Solution 4 − 48 2 Rewrite the radicand as a product using the largest perfect square factor. 4 − 16 · 3 2 Rewrite the radical as the product of two radicals. 4 − 16 · 3 2 Simplify. 4 − 4 3 2 Factor the common factor from the numerator. 4 ⎛ ⎝ 1 − 3 ⎞ ⎠ 2 Remove the common factor, 2, from the numerator and denominator. 2 · 2 ⎛ ⎝ 1 − 3 ⎞ ⎠ 2 Simplify. 2 ⎛ ⎝ 1 − 3 ⎞ ⎠ TRY IT : : 9.37 Simplify: 10 − 75 5 . TRY IT : : 9.38 Simplify: 6 − 45 3 . Use the Quotient Property to Simplify Square Roots Whenever you have to simplify a square root, the first step you should take is to determine whether the radicand is a perfect square. A perfect square fraction is a fraction in which both the numerator and the denominator are perfect squares. EXAMPLE 9.20 Simplify: 9 64 . Solution 9 64 Since ⎛ ⎝ 3 8 ⎞ ⎠ 2 = 9 64 3 8 TRY IT : : 9.39 Simplify: 25 16 . TRY IT : : 9.40 Simplify: 49 81 . Chapter 9 Roots and Radicals 1047 If the numerator and denominator have any common factors, remove them. You may find a perfect square fraction! EXAMPLE 9.21 Simplify: 45 80 . Solution 45 80 Simplify inside the radical first. Rewrite showing the common factors of the numerator and denominator. 5 · 9 5 · 16 Simplify the fraction by removing common factors. 9 16 Simplify. ⎛ ⎝ 3 4 ⎞ ⎠ 2 = 9 16 3 4 TRY IT : : 9.41 Simplify: 75 48 . TRY IT : : 9.42 Simplify: 98 162 . In the last example, our first step was to simplify the fraction under the radical by removing common factors.
eBook - PDF- Jerome Kaufmann, Karen Schwitters, , , Jerome Kaufmann, Karen Schwitters(Authors)
- 2014(Publication Date)
- Cengage Learning EMEA(Publisher)
5.5 • Equations Involving Radicals 273 ✔ Check y 1 6 5 y y 1 6 5 y 4 1 6 0 4 when y 5 4 or 9 1 6 0 9 when y 5 9 2 1 6 0 4 3 1 6 0 9 8 ? 4 9 5 9 The only solution is 9 ; the solution set is 5 9 6 . In Example 4, note that we changed the form of the original equation y 1 6 5 y to y 5 y 2 6 before we squared both sides. Squaring both sides of y 1 6 5 y produces y 1 12 y 1 36 5 y 2 , which is a much more complex equation that still contains a radical. Here again, it pays to think ahead before carrying out all the steps. Now let’s consider an example involving a cube root. Solve 3 n 2 2 1 5 2 . Solution 3 n 2 2 1 5 2 Q 3 n 2 2 1 R 3 5 2 3 Cube both sides n 2 2 1 5 8 n 2 2 9 5 0 ( n 1 3)( n 2 3) 5 0 n 1 3 5 0 or n 2 3 5 0 n 5 2 3 or n 5 3 ✔ Check 3 n 2 2 1 5 2 3 n 2 2 1 5 2 3 ( 2 3) 2 2 1 0 2 when n 5 2 3 or 3 3 2 2 1 0 2 when n 5 3 3 8 0 2 3 8 0 2 2 5 2 2 5 2 The solution set is 5 2 3, 3 6 . It may be necessary to square both sides of an equation, simplify the resulting equation, and then square both sides again. The next example illustrates this type of problem. Solve x 1 2 5 7 2 x 1 9 . Solution x 1 2 5 7 2 x 1 9 Q x 1 2 R 2 5 Q 7 2 x 1 9 R 2 Square both sides x 1 2 5 49 2 14 x 1 9 1 x 1 9 x 1 2 5 x 1 58 2 14 x 1 9 2 56 5 2 14 x 1 9 4 5 x 1 9 (4) 2 5 Q x 1 9 R 2 Square both sides 16 5 x 1 9 7 5 x EXAMPLE 5 Classroom Example Solve 3 x 2 1 2 5 3 . EXAMPLE 6 Classroom Example Solve x 1 4 5 1 1 x 2 1 . Copyright 2013 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.
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