Powers Roots And Radicals

8 Key excerpts on "Powers Roots And Radicals"

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  Intermediate Algebra

  • Mark D. Turner, Charles P. McKeague(Authors)
  • 2016(Publication Date)
  • XYZ Textbooks

    (Publisher)

  Chapter Outline 7.1 Roots and Radical Functions 7.2 Rational Exponents 7.3 Simplified Form for Radicals 7.4 Addition and Subtraction of Radical Expressions 7.5 Multiplication and Division of Radical Expressions 7.6 Equations Involving Radicals 7.7 Complex Numbers 7 iStockphoto.com © trait2lumiere 489 E cology and conservation are topics that interest most college students. If our rivers and oceans are to be preserved for future generations, we need to work to eliminate pollution from our waters. If a river is flowing at 1 meter per second and a pollutant is entering the river at a constant rate, the shape of the pollution plume can often be modeled by the simple equation y = √ — x The following table and graph were produced from the equation. Distance from Width of Source (meters) Plume (meters) x y 0 0 1 1 4 2 9 3 16 4 Width of a Pollutant Plume 4 0 8 12 16 20 4 0 8 12 16 20 x y Distance from source (m) Width of plume (m) To visualize how the graph models the pollutant plume, imagine that the river is flowing from left to right, parallel to the x -axis, with the x -axis as one of its banks. The pollutant is entering the river from the bank at (0, 0). By modeling pollution with mathematics, we can use our knowledge of mathematics to help control and eliminate pollution. Roots and Rational Exponents 490 Success Skills If you have made it this far, then you have the study skills necessary to be successful in this course. Success skills are more general in nature and will help you with all your classes and ensure your success in college as well. Let's start with a question: Question: What quality is most important for success in any college course? Answer: Independence. You want to become an independent learner. We all know people like this. They are generally happy. They don't worry about getting the right instructor, or whether or not things work out every time.

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  Elementary and Intermediate Algebra

  • Mark D. Turner, Charles P. McKeague(Authors)
  • 2016(Publication Date)
  • XYZ Textbooks

    (Publisher)

  Chapter Outline 10.1 Roots and Radical Functions 10.2 Rational Exponents 10.3 Simplified Form for Radicals 10.4 Addition and Subtraction of Radical Expressions 10.5 Multiplication and Division of Radical Expressions 10.6 Equations Involving Radicals 10.7 Complex Numbers 10 iStockphoto.com © trait2lumiere 699 E cology and conservation are topics that interest most college students. If our rivers and oceans are to be preserved for future generations, we need to work to eliminate pollution from our waters. If a river is flowing at 1 meter per second and a pollutant is entering the river at a constant rate, the shape of the pollution plume can often be modeled by the simple equation y = √ — x The following table and graph were produced from the equation. Distance from Width of Source (meters) Plume (meters) x y 0 0 1 1 4 2 9 3 16 4 Width of a Pollutant Plume 4 0 8 12 16 20 4 0 8 12 16 20 x y Distance from source (m) Width of plume (m) To visualize how the graph models the pollutant plume, imagine that the river is flowing from left to right, parallel to the x -axis, with the x -axis as one of its banks. The pollutant is entering the river from the bank at (0, 0). By modeling pollution with mathematics, we can use our knowledge of mathematics to help control and eliminate pollution. Roots and Rational Exponents 700 Success Skills Think about the most successful people you have met or heard about. What are the qualities they tend to have in common? One of these qualities usually involves making a resolute commitment. If you are not firmly committed to something, then you will tend to give less than your full effort. Consider this quote from Faust by Johann Wolfgang Von Goethe: Until one is committed, there is hesitancy, the chance to draw back, always ineffectiveness.

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  eBook - ePub

  HP Prime Guide Algebra Fundamentals

  HP Prime Revealed and Extended

  • Larry S Schroeder(Author)
  • 2017(Publication Date)
  • Larry Schroeder

    (Publisher)

  Explanation 1.4 – Radicals and Rational Exponents

  In this section we introduce radicals and rational exponents. We start by going over the difference between the square root of a number and the principal square root. We expand this to the nth root and the principal nth root.

  We then use the radicals to define rational exponents. The rational exponents are also referred to as fractional exponents. It can be shown from the Definition of Rational Exponents that the Properties of Exponents hold as well.

  We conclude this section with eliminating radicals in the denominator. This process is referred to as rationalizing the denominator.

  Radicals and Their Properties

  A number is squared when it is raised to the second power. Many times we need to know what number was squared to produce a value of a. If this value exist we refer to that number as a square root of a.

  Thus

  25 has -5 and 5 as square roots since (-5)2 = 25 and (5)2 = 25,

  49 has -7 and 7 as square roots since (-7)2 = 49 and (7)2 = 49,

  -16 has no real number square root since no real number b where b2 = -16.

  Zero only has itself as a square root. We will later add the complex number system where square roots exist for negative numbers.

  HP Prime Family Square Root - solve

  Begin by selecting the CAS key on the HP Prime. If the CAS view of the screenshot has computations, clear the history first. To clear the history, press the Clear key.

  Key in as shown. Use the Toobox key to enter solve() . Select Toolbox > CAS > Solve > Solve

  and press Enter

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  GRE All the Quant

  Effective Strategies & Practice from 99th Percentile Instructors

  • (Author)
  • 2023(Publication Date)
  • Manhattan Prep

    (Publisher)

  CHAPTER 5 Exponents and Roots

  So far, exponents have been defined as a shorthand way of expressing multiplication. For example, 52 = 5 × 5 = 25 and because 92 = 81. For larger numbers, however, this approach could be prohibitively time consuming, and it’s all but impossible when you have variables. In this chapter, you’ll learn all of the exponent and root rules that will allow you to combine exponential terms and simplify complex expressions.

  Exponents and Roots Language

  Have you ever heard the expression, “Wow, that increased exponentially!”? This expression captures the essence of exponents. When a number greater than 1 increases exponentially, it does not merely increase; it increases a significant amount and it does so very rapidly.

  In fact, the greater the exponent, the faster the rate of increase. Consider the following progression:

  This trend holds true when positive bases greater than 1 are raised to higher and higher powers. With many other numbers, though, this trend will not necessarily hold true. For example, when the number 1 is raised to any exponent, it does not increase at all; it remains 1.

  The expression 43 consists of a base (4) and an exponent (3). This expression is read as “four to the third power” or “four cubed” and means four multiplied by itself three times. Thus, four cubed is 43 = 4 × 4 × 4 = 64.

  Roots undo exponents. Asking for the cube root of 64 is the same thing as asking “What number, when cubed, gives 64?” Thus, . Four cubed is 64, and 64 cube rooted is 4.

  Most exponents will be expressed as “the base (raised) to the power of the exponent.” So 35 is called “three to the fifth power” (and equals 243, incidentally). To undo that, you would take the fifth root of 243, which is written as and which equals 3.

  Something raised to the second power is called a square, and something raised to the third power is a cube. After that, use the number of the power (fourth power, fifth power, sixth power). For second and third powers, the GRE may use either the special names (square, cube) or the more traditional ones.

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  Intermediate Algebra

  • Jerome Kaufmann, Karen Schwitters, , , Jerome Kaufmann, Karen Schwitters(Authors)
  • 2014(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  One possibility is that you made a mistake in working the problem. Another possibility is that the answer in the back of the book is incorrect. It could be that both your version of the answer and the answer in the back of the book are equivalent forms and hence both are correct answers. If you think that is the case, be sure to ask the instructor. Exponents and Radicals © l i g h t p o e t/Shutterstock.com 5.1 Using Integers as Exponents 5.2 Roots and Radicals 5.3 Combining Radicals and Simplifying Radicals That Contain Variables 5.4 Products and Quotients Involving Radicals 5.5 Equations Involving Radicals 5.6 Merging Exponents and Roots 5.7 Scientific Notation “One of life’s most painful moments comes when we must admit that we didn’t do our homework, that we are not prepared.” merlin olsen, nfl player and actor What is your plan for being sure you have time to do your math homework? 243 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. Chapter 5 • Exponents and Radicals 244 Thus far in the text we have used only positive integers as exponents. In Chapter 1 the expres-sion b n , where b is any real number and n is a positive integer, was defined by b n 5 b ? b ? b ? # # # ? b n factors of b Then, in Chapter 3, more properties of exponents served as a basis for manipulation with polynomials. We are now ready to extend the concept of an exponent to include the use of zero and the negative integers as exponents. First, let’s consider the use of zero as an exponent. We want to use zero in such a way that the previously listed properties continue to hold.

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  Intermediate Algebra

  Concepts and Graphs 2E

  • Charles P. McKeague(Author)
  • 2019(Publication Date)
  • XYZ Textbooks

    (Publisher)

  496 CHAPTER 6 Rational Exponents and Roots EXAMPLE Assume all variables represent nonnegative numbers, and simplify each expression as much as possible. 7. √ — 25a 4 b 6 = 5a 2 b 3 because (5a 2 b 3 ) 2 = 25a 4 b 6 . 8. 3 √ — x 6 y 12 = x 2 y 4 because (x 2 y 4 ) 3 = x 6 y 12 . 9. 4 √ — 81r 8 s 20 = 3r 2 s 5 because (3r 2 s 5 ) 4 = 81r 8 s 20 . Rational Numbers as Exponents We will now develop a second kind of notation involving exponents that will allow us to designate square roots, cube roots, and so on in another way. Consider the equation x = 8 1/3 . Although we have not encountered fractional exponents before, let’s assume that all the properties of exponents hold in this case. Cubing both sides of the equation, we have x 3 = (8 1/3 ) 3 x 3 = 8 (1/3)(3) x 3 = 8 1 x 3 = 8 The last line tells us that x is the number whose cube is 8. It must be true, then, that x is the cube root of 8, x = 3 √ — 8 . Because we started with x = 8 1/3 , it follows that 8 1/3 = 3 √ — 8 It seems reasonable, then, to define fractional exponents as indicating roots. Here is the formal definition. With this definition, we have a way of representing roots with exponents. Here are some examples. EXAMPLE Write each expression as a root and then simplify, if possible. 10. 8 1/3 = 3 √ — 8 = 2 11. 36 1/2 = √ — 36 = 6 12. −25 1/2 = − √ — 25 = −5 13. (−25) 1/2 = √ — −25 , which is not a real number 14.  4 __ 9  1/2 = √ __ 4 __ 9 = 2 __ 3 C DEFINITION If x is a real number and n is a positive integer greater than 1, then x 1/n = n √ — x (x ≥ 0 when n is even) In words: The quantity x 1/n is the nth root of x. 6.1 Rational Exponents 497 The properties of exponents developed in Chapter 1 were applied to integer exponents only. We will now extend these properties to include rational exponents also. We do so without proof. Sometimes rational exponents can simplify our work with radicals. Here are Examples 8 and 9 again, but this time we will work them using rational exponents.

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  Elementary Algebra

  • Alan Tussy, R. Gustafson(Authors)
  • 2012(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  By definition, if . b 3 a 2 3 a b 2 3 a a because . 4 3 64 2 3 64 4 Just as there are square roots and cube roots, there are also fourth roots, fifth roots, and so on. The th root of is written , and if . The number is called the index of the radical. n b n a 2 n a b 2 n a a n The index is 4. The index is 5. 2 5 32 m 10 2 m 2 because ( 2 m 2 ) 5 32 m 10 2 4 81 3 because 3 4 81 When is even, we say that the radical is an even root. When is odd, we say that the radical is an odd root. 2 n a n 2 n a n is an even root. is not a real number. No real number raised to the fourth power is . is an odd root. 2 3 125 12 2 4 12 2 4 16 Copyright 201 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. 670 CHAPTER 8 Radical Expressions and Equations The product and quotient rules for square roots can be generalized to apply to higher-order roots. They can be used to simplify radical expressions. B n a b 2 n a 2 n b 1 2 n b 0 2 2 n a b 2 n a 2 n b Simplify: Write as a product of its greatest perfect fourth power factor, 16, and one other factor. The fourth root of a product is equal to the product of the fourth roots. Evaluate: . Simplify: The cube root of a quotient is equal to the quotient of the cube roots. In the denominator, evaluate: . 2 3 27 3 2 3 26 3 B 3 26 27 2 3 26 2 3 27 2 4 16 2 2 2 4 2 a 2 4 16 2 4 2 a 32 a 2 4 32 a 2 4 16 2 a To evaluate exponential expressions involving fractional exponents, use the rules for rational exponents to write the expressions in an equivalent radical form.

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  Introductory 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.

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