Powers and Roots

6 Key excerpts on "Powers and Roots"

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  eBook - ePub

  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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  eBook - ePub

  Foundations of GMAT Math

  • Manhattan GMAT(Author)
  • 2011(Publication Date)
  • Manhattan Prep Publishing

    (Publisher)

  Chapter 3: Exponents & Roots  

  In This Chapter:

  •  Rules of exponents •  Rules of roots Basics of Exponents

  To review, exponents represent repeated multiplication. The exponent, or power, tells you how many bases to multiply together.

  53	=	5 × 5 × 5	=	125
  Five cubed	equals	three fives multiplied together, or five times five times five,	which equals	one hundred twenty-five.

  An exponential expression or term simply has an exponent in it. Exponential expressions can contain variables as well. The variable can be the base, the exponent, or even both.

  a4	=	a × a × a × a
  a to thefourth	equals	four a's multipliedtogether, or a times atimes a times a.

  3 x	=	3 × 3 ×…× 3
  Three to thexth power	equals	three times three timesdot dot dot times three.There are x three's in theproduct, whatever x is.

  Any base to the first power is just that base.

  71	=	7
  Seven to the first	equals	seven.

  Memorize the following powers of positive integers.

  Squares	Cubes
  12 = 122 = 432 = 942 = 1652 = 2562 = 3672 = 4982 = 6492 = 81102 = 100112 = 121122 = 144132 = 169142 = 196152 = 225202 = 400302 = 900	13 = 123 = 833 = 2743 = 6453 = 125103 = 1,000 Powers of 2 21 = 222 = 423 = 824 = 1625 = 3226 = 6427 = 12828 = 25629 = 512210 = 1,024

  Powers of 3 31 = 332 = 933 = 2734 = 81

  Powers of 4 41 = 442 = 1643 = 64

  Powers of 5 51 = 552 = 2553 = 125

  Powers of 10 101 = 10102 = 100103 = 1,000

  Remember PEMDAS? Exponents come before everything else, except Parentheses. That includes negative signs.

  –32	=	–(32 )	=	–9
  The negative of three squared	equals	the negative of the quantity three squared,	which equals	negative nine.

  To calculate –32 , square the 3 before you multiply by negative one (–1). If you want to square the negative sign, throw parentheses around –3.

  (–3)2	=	9
  The square ofnegative three	equals	nine.

  In (–3)2 , the negative sign and the three are both inside the parentheses, so they both get squared. If you say “negative three squared,” you probably mean (–3)2 , but someone listening might write down –32

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  eBook - PDF

  College Algebra

  Building Skills and Modeling Situations

  • Charles P. McKeague, Katherine Yoshiwara, Denny Burzynski(Authors)
  • 2013(Publication Date)
  • XYZ Textbooks

    (Publisher)

  EXAMPLES EXAMPLES 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. DEFINITION 1.6 Roots and Radicals 51 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 pos- sible. 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 The properties of exponents developed earlier were applied to integer exponents only. We will now extend these properties to include rational exponents also. We do so without proof. Here are Examples 8 and 9 again, but this time we will work them using rational exponents. Write each radical with a rational exponent, then simplify. 15. 3 √ — x 6 y 12 = (x 6 y 12 ) 1/3 = (x 6 ) 1/3 (y 12 ) 1/3 = x 2 y 4 16. 4 √ — 81r 8 s 20 = (81r 8 s 20 ) 1/4 = 81 1/4 (r 8 ) 1/4 (s 20 ) 1/4 = 3r 2 s 5 EXAMPLES If a and b are real numbers and r and s are rational numbers, and a and b are nonnegative whenever r and s indicate even roots, then 1. a r ⋅ a s = a r+s 4. a −r = 1 __ a r (a ≠ 0) 2. (a r ) s = a rs 5.  a __ b  r = a r __ b r (b ≠ 0) 3. (ab ) r = a r b r 6. a r __ a s = a r−s (a ≠ 0) PROPERTY Properties of Exponents EXAMPLES 52 Chapter 1 Algebra Review 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. Proof We can prove our theorem using the properties of exponents. Because m/n = m(1/n), we have a m/n = a m(1/n) a m/n = a (1/n)(m) = (a m ) 1/n = (a 1/n ) m Here are some examples that illustrate how we use this theorem. Simplify as much as possible. 17. 8 2/3 = (8 1/3 ) 2 Rational Exponents Theorem = 2 2 Definition of rational exponents = 4 The square of 2 is 4 18.

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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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  Intermediate Algebra

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

    (Publisher)

  The definition of the cube root of a number is as follows. Definition 5.4 Cube Root of a Number 3 b 5 a if and only if a 3 5 b . The symbol 3 designates the cube root of a number. The following examples sum-marize some of the information about cube roots. 3 125 5 5 Five is the cube root of 125 because 5 3 5 125 . The cube root of a positive number is always a positive number. 3 2 27 5 2 3 Negative three is the cube root of 2 27 because ( 2 3) 3 5 2 27 . The cube root of a negative number is always a negative number. 3 0 5 0 The cube root of zero is zero. Remark: Technically, every nonzero real number has three cube roots, but only one of them is a real number. The other two roots are classified as imaginary numbers. We are restricting our work at this time to the set of real numbers. The concept of root can be extended to fourth roots, fifth roots, sixth roots, and in gen-eral, n th roots. The fourth root of a number is one of its four equal factors. Thus 3 is a fourth root of 81 because 3 4 5 81 . The definition of the n th root of a number is as follows. Definition 5.5 n th Root of a Number The n th root of b is a if and only if a n 5 b . The symbol n designates the principal n th root. To complete our terminology, the n in the radical n b is called the index of the radical . For square roots, when n 5 2 , we com-monly write b rather than 2 b . The following examples summarize some of the informa-tion about higher power roots. 4 16 5 2 4 16 indicates the principal fourth root of 16, which is 2 because 2 4 5 16 . 5 2 32 5 2 2 The principal fifth root of 2 32 is 2 2 because ( 2 2) 5 5 2 32. 4 2 81 The fourth root of 2 81 does not exist in the real number system be-cause there is no real number that when raised to the fourth power gives a negative value. Because the rules are different for even-numbered roots and odd-numbered roots, let’s make some generalizations about roots for each category. Copyright 2013 Cengage Learning. All Rights Reserved.

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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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