Mathematics
Rational Exponents
Rational exponents are exponents that are expressed as fractions. They allow for the calculation of roots and powers of numbers using fractional exponents, providing a way to represent radical expressions in a more convenient form. Rational exponents are a fundamental concept in algebra and are used to simplify and manipulate expressions involving roots and powers.
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eBook - PDFIntermediate 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.
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
Here is the formal definition. With this definition, we have a way of representing roots with exponents. Here are some examples. Write each expression as a root and then simplify, if possible. 1. 8 1/3 = 3 √ — 8 = 2 2. 36 1/2 = √ — 36 = 6 3. − 25 1/2 = − √ — 25 = − 5 4. ( − 25) 1/2 = √ — − 25 , which is not a real number 5. 4 __ 9 1/2 = √ __ 4 __ 9 = 2 __ 3 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 n th root of x . DEFINITION VIDEO EXAMPLES SECTION 7.2 EXAMPLES 502 CHAPTER 7 Roots and Rational Exponents 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 12 and 13 from Section 7.1 again, but this time we will work them using Rational Exponents. Write each radical with a rational exponent, then simplify. 6. 3 √ — x 6 y 12 = ( x 6 y 12 ) 1/3 = ( x 6 ) 1/3 ( y 12 ) 1/3 = x 2 y 4 7. 4 √ — 81 r 8 s 20 = (81 r 8 s 20 ) 1/4 = 81 1/4 ( r 8 ) 1/4 ( s 20 ) 1/4 = 3 r 2 s 5 So far, the numerators of all the Rational Exponents we have encountered have been 1. The next theorem extends the work we can do with Rational Exponents to Rational Exponents with numerators other than 1. We can extend our properties of exponents with the following theorem. If a is a nonnegative real number, m is an integer, and n is a positive integer, then a m / n = a 1/ n m = n √ — a m and a m / n = ( a m ) 1/ n = n √ — a m With Rational Exponents, the numerator always represents a power and the denominator represents the index of a root. Theorem 7.1 Proof We can prove Theorem 7.1 using the properties of exponents.
eBook - PDF- Mark D. Turner, Charles P. McKeague(Authors)
- 2016(Publication Date)
- XYZ Textbooks(Publisher)
Here is the formal definition. With this definition, we have a way of representing roots with exponents. Here are some examples. Write each expression as a root and then simplify, if possible. 1. 8 1/3 = 3 √ — 8 = 2 2. 36 1/2 = √ — 36 = 6 3. − 25 1/2 = − √ — 25 = − 5 4. ( − 25) 1/2 = √ — − 25 , which is not a real number 5. 4 __ 9 1/2 = √ __ 4 __ 9 = 2 __ 3 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 n th root of x . DEFINITION VIDEO EXAMPLES SECTION 10.2 EXAMPLES 712 Chapter 10 Roots and Rational Exponents 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 12 and 13 from Section 10.1 again, but this time we will work them using Rational Exponents. Write each radical with a rational exponent, then simplify. 6. 3 √ — x 6 y 12 = ( x 6 y 12 ) 1/3 = ( x 6 ) 1/3 ( y 12 ) 1/3 = x 2 y 4 7. 4 √ — 81 r 8 s 20 = (81 r 8 s 20 ) 1/4 = 81 1/4 ( r 8 ) 1/4 ( s 20 ) 1/4 = 3 r 2 s 5 So far, the numerators of all the Rational Exponents we have encountered have been 1. The next theorem extends the work we can do with Rational Exponents to Rational Exponents with numerators other than 1. We can extend our properties of exponents with the following theorem. If a is a nonnegative real number, m is an integer, and n is a positive integer, then a m / n = a 1/ n m = n √ — a m and a m / n = ( a m ) 1/ n = n √ — a m With Rational Exponents, the numerator always represents a power and the denominator represents the index of a root. THEOREM Theorem 7.1 Proof We can prove Theorem 7.1 using the properties of exponents.
eBook - PDF- Michael A. Calter, Paul A. Calter, Paul Wraight, Sarah White(Authors)
- 2016(Publication Date)
- Wiley(Publisher)
13–1 Integral Exponents In this section, we continue the study of exponents that we started in Sec. 2–3. We repeat the laws of exponents that were explained there and use them to simplify harder expressions than before. You should refresh your memory on the laws of exponents covered in that section before going too far into this chapter. 13 ◆◆◆ OBJECTIVES ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ When you have completed this chapter, you should be able to: • Apply the laws of exponents to simplify expressions; • Apply the laws of radicals to simplify them into standard form; • Perform operations with radicals including addition, subtraction, multiplication, and division; • Solve equations containing radicals. In Chapter 2, we learned: • How to raise a variable to a power; and • How to take a root. In Chapter 13, we will now take what we learned from Chapter 2 and apply it to entire expres- sions or the entire side of an equation. You will need the information learned in this chapter to bring expressions out from under a root sign, or remove the root signs altogether. Many students ask, “Why do I need to learn to do this by hand when my calculator can do this for me?” Our answer is simply: 1. calculators are quirky and very sensitive to syntax; the way an equation or expression is entered into the calculator can make a big difference. Sometimes it’s not enough to enter it into the calculator even though it is mathematically correct; and 2. experts in science and engineering technology should not be at the sole mercy of their calculators. Exponents and Radicals
eBook - PDF- Paul A. Calter, Michael A. Calter(Authors)
- 2011(Publication Date)
- Wiley(Publisher)
379 Exponents and Radicals ◆◆◆ OBJECTIVES ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ When you have completed this chapter, you should be able to • Use the laws of exponents to simplify and combine expressions having integral exponents, by hand or by calculator. • Simplify radicals by removing perfect powers, by rationalizing the denominator, and by reducing the index. • Add, subtract, multiply, and divide radicals. • Solve radical equations, manually or by calculator. ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ We introduced exponents and gave the laws of exponents in our “Introduction to Algebra” chapter. We review those laws here, give more advanced examples of their use, with applications, and show how to manipulate expressions having exponents by calculator. Next we make a strong connection between exponents and radicals, and show how to simplify, add, subtract, multiply, and divide radical expressions. We did some calculation of roots in Chapter 1, but with only numbers under the radical sign. Here we show how to handle expressions with literals under the radical sign. The ability to manipulate both exponents and radicals is needed to work with many formulas found in technology. Finally we add another kind of equation to our growing list, the radical equation. As with quadratics, we start with methods of solution that we already know, solu- tion by graphing and by calculator. This is followed by methods for an algebraic so- lution, and of course, applications. For example, the natural frequency of the weight bouncing at the end of a spring, Fig. 13–1, is given by where g is the gravitational constant, k is the spring constant, and W is the weight. You would have no problem finding , given the other quantities, but how would you solve for, say, W? You will learn how in this chapter. f n f n 1 2pA kg W f n 13 W FIGURE 13–1 380 Chapter 13 ◆ Exponents and Radicals TI-89 screen for Example 2.
eBook - PDFAlgebra
A Combined Course 2E
- Charles P. McKeague(Author)
- 2018(Publication Date)
- XYZ Textbooks(Publisher)
GETTING READY FOR CLASS After reading through the preceding section, respond in your own words and in complete sentences. A. What is an exponent? B. In symbols, show that any number other than zero raised to the 0 power is 1. C. Briefly list the steps of the order of operations. D. Find the value of the expression 3(4t − 2) if t = 2. Answers 12. 16 13. 0, 18, 0 12. Find the value of 5x + 2y − 10 when x is 4 and y is 3. 13. Evaluate (x + 3) 2 , x 2 + 9, and x 2 + 6x + 9 when x is −3. a (a + 4) 2 a 2 + 16 a 2 + 8a + 16 −2 (−2 + 4) 2 = 4 (−2) 2 + 16 = 20 (−2) 2 + 8(−2) + 16 = 4 0 (0 + 4) 2 = 16 0 2 + 16 = 16 0 2 + 8(0) + 16 = 16 3 (3 + 4) 2 = 49 3 2 + 16 = 25 3 2 + 8(3) + 16 = 49 82 Chapter 1 Real Numbers and Algebraic Expressions E X E R C I S E S E T 1.2 VOCABULARY REVIEW Choose the correct words to fill in the blanks below. zero first base product difference factor sum exponent quotient 1. For the exponential expression 5 3 , the 5 is called the and the 3 is called the . 2. An exponent is a number that indicates how many times the base is used as a . 3. Any number raised to the power is the number itself. 4. Any non-zero number raised to the power is 1. 5. Twice the of 6 and 1 is written in symbols as 2(6 + 1). 6. 4 added to 5 times the of 3 and 2 is written in symbols as 5(3 − 2) + 4. 7. 7 subtracted from the of 4 and 3 is written in symbols as 4 ∙ 3 − 7. 8. The of 8 and 2 plus 9 is written in symbols as 8 ÷ 2 + 9. The following is a list of steps to perform operations when evaluating mathematical expressions. Number the steps in the correct order. Do all multiplications and divisions in order, left to right. Simplify any numbers with exponents. Perform operations inside the grouping symbols, or above and below the fraction bar. Do all additions and subtractions, in order left to right. A For each of the following expressions, name the base and the exponent. 1. 4 5 2. 5 4 3. 3 6 4. 6 3 5. 8 2 6. 2 8 7. 9 1 8. 1 9 9.
eBook - PDF- Alan Tussy, R. Gustafson(Authors)
- 2012(Publication Date)
- Cengage Learning EMEA(Publisher)
To avoid large numbers, we usually find the root of the base first and then calculate the power using the relationship . a m > n A 1 n a B m m n ( a m ) 1 > n 2 n a m ( a 1 > n ) m A 1 n a B m ( a m ) 1 > n ( a 1 > n ) m a m > n 4 3 > 2 (4 3 ) 1 > 2 64 1 > 2 2 64 8 (4 3 ) 1 > 2 4 3 > 2 4 3 > 2 4 3 > 2 (4 1 > 2 ) 3 1 2 4 2 3 2 3 8 (4 1 > 2 ) 3 4 3 > 2 a 1 > n a m > n 8.6 Higher-Order Roots and Rational Exponents 661 If and represent positive integers and represents a real number, and a m > n 2 n a m a m > n A 1 n a B m 1 n a ( n 1) n m Definition of a m / n EXAMPLE 7 Evaluate: a. b. c. d. Strategy We will identify the base and the exponent of the exponential expression so that we can write the exponential expression in an equivalent radical form. Why We know how to evaluate square roots, cube roots, and fourth roots. Solution a. In the exponential expression , the base is 125 and the exponent is . The base is the same as the radicand of the corresponding radical. The denominator of the rational exponent is the same as the radical’s index. The numerator of the rational exponent indicates the power to which the radical base is raised. Power Root Read as “125 to the four-thirds power.” Because the exponent is , find the cube root of the base, 125, to get 5. Then find the fourth power of 5. Base b. Because the exponent is , find the fourth root of the base, 81, to get 3. Then find the third power of 3. c. For the exponential expression , the base is 25, not . Because the exponent is , find the square root of the base, 25, to get 5. Then find the third power of 5. d. Because of the parentheses, the base of the exponential expression is . Because the exponent is , find the cube root of the base, , to get .
eBook - PDF- Sheldon Axler(Author)
- 2011(Publication Date)
- Wiley(Publisher)
chapter 4 Polynomial and Rational Functions We begin this chapter by reviewing the properties of integer exponents, in Statue of the Persian mathematician and poet Omar Khayyam, whose algebra book written in 1070 con- tained the first seri- ous study of cubic polynomials. preparation for dealing with polynomial functions. We will see why x 0 is defined to be 1 and why x -m is defined to be 1 x m . Then we deal with polynomials, one of the most important classes of functions. We will look at the connection between the zeros of a polynomial and its linear factors. We will also examine the behavior of the graphs of polynomials. Next we turn to rational functions, which are ratios of polynomials. Unlike polynomials, rational functions can have asymptotes in their graphs. This chapter concludes with a section on complex numbers, including a discussion of complex zeros of polynomials. 213 214 chapter 4 Polynomial and Rational Functions 4.1 Integer Exponents learning objectives By the end of this section you should be able to explain why x 0 is defined to equal 1 (for x 6= 0); explain why x -m is defined to equal 1 x m (for m a positive integer and x 6= 0); manipulate and simplify expressions involving integer exponents. Positive Integer Exponents Multiplication by a positive integer is repeated addition, in the sense that if x is a real number and m is a positive integer, then mx equals the sum with x appearing m times: mx = x + x + · · · + x | {z } x appears m times . Just as multiplication by a positive integer is defined as repeated addition, positive integer exponents denote repeated multiplication: Positive integer exponents If x is a real number and m is a positive integer, then x m is defined to be the product with x appearing m times: x m = x · x · · · · · x | {z } x appears m times . example 1 Evaluate ( 1 2 ) 3 . solution ( 1 2 ) 3 = 1 2 · 1 2 · 1 2 = 1 8 If m is a positive integer, then we can define a function f by f (x) = x m .
eBook - PDF- Sheldon Axler(Author)
- 2011(Publication Date)
- Wiley(Publisher)
chapter 4 Polynomial and Rational Functions We begin this chapter by reviewing the properties of integer exponents, in Statue of the Persian mathematician and poet Omar Khayyam, whose algebra book written in 1070 con- tained the first seri- ous study of cubic polynomials. preparation for dealing with polynomial functions. We will see why x 0 is defined to be 1 and why x -m is defined to be 1 x m . Then we deal with polynomials, one of the most important classes of functions. We will look at the connection between the zeros of a polynomial and its linear factors. We will also examine the behavior of the graphs of polynomials. Next we turn to rational functions, which are ratios of polynomials. Unlike polynomials, rational functions can have asymptotes in their graphs. This chapter concludes with a section on complex numbers, including a discussion of complex zeros of polynomials. 213 214 chapter 4 Polynomial and Rational Functions 4.1 Integer Exponents learning objectives By the end of this section you should be able to explain why x 0 is defined to equal 1 (for x = 0); explain why x -m is defined to equal 1 x m (for m a positive integer and x = 0); manipulate and simplify expressions involving integer exponents. Positive Integer Exponents Multiplication by a positive integer is repeated addition, in the sense that if x is a real number and m is a positive integer, then mx equals the sum with x appearing m times: mx = x + x + · · · + x x appears m times . Just as multiplication by a positive integer is defined as repeated addition, positive integer exponents denote repeated multiplication: Positive integer exponents If x is a real number and m is a positive integer, then x m is defined to be the product with x appearing m times: x m = x · x · · · · · x x appears m times . example 1 Evaluate ( 1 2 ) 3 . solution ( 1 2 ) 3 = 1 2 · 1 2 · 1 2 = 1 8 If m is a positive integer, then we can define a function f by f (x) = x m .
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)
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. 9.4 Simplifying and Combining Radical Expressions 675 Simplify a radical expression by applying the properties of radicals. Many properties of exponents have counterparts in radical notation. Because a 1/ n b 1/ n 5 1 ab 2 1/ n , we have n a n b 5 n ab For example, 5 5 5 5 ? 5 5 5 2 5 5 3 7 x 3 49 x 2 5 3 7 x ? 7 2 x 2 5 3 7 3 ? x 3 5 7 x 4 2 x 3 4 8 x 5 4 2 x 3 ? 2 3 x 5 4 2 4 ? x 4 5 2 x 1 x . 0 2 If we write Equation 1 in a different order, we have the following rule. 1 (1) MULTIPLICATION PROPERTY OF RADICALS If n a and n b are real numbers, then n ab 5 n a n b If all radicals represent real numbers, the nth root of the product of two numbers is equal to the product of their nth roots . Comment The multiplication property of radicals applies to the n th root of the product of two numbers. There is no such property for sums or differences. A radical symbol is a grouping symbol. Thus, any addition or subtraction within the radicand must be completed first. A second property of radicals involves quotients. Because a 1/ n b 1/ n 5 a a b b 1/ n 1 b 2 0 2 it follows that ! n a n b 5 Å n a b 1 b u 0 2 For example, 8 x 3 2 x 5 Å 8 x 3 2 x 5 4 x 2 5 2 x 1 x . 0 2 3 54 x 5 3 2 x 2 5 Å 3 54 x 5 2 x 2 5 3 27 x 3 5 3 x 1 x 2 0 2 If we write Equation 2 in a different order, we have the following rule. Carl Friedrich Gauss 1777–1855 Many people consider Gauss to be the greatest mathematician of all time. He made contributions in the areas of number theory, solutions of equations, geometry of curved surfaces, and statistics.
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