Rational Exponents

3 Key excerpts on "Rational Exponents"

• 

  eBook - ePub

  Intermediate Algebra

  • Lisa Healey(Author)
  • 2021(Publication Date)
  • Chemeketa Press

    (Publisher)

  B. Definition of Rational Exponents

  We can also have fractional exponents with numerators other than 1. The exponent must be a fraction in lowest terms. Because a fraction is a ratio between two numbers, we refer to all fractional exponents as

  Rational Exponents

  .

  The product property of integer exponents from Section 2.1 works for Rational Exponents. Using the product property and the fact , we can write expressions with Rational Exponents in two ways. Because the exponent m/n = m(1/n) = (1/n)m we have

  and

  Definition of Rational Exponents

  If m is an integer, n is a natural number, and is a real number, then

  A power of the form is said to have a

  rational exponent

  .

  The numerator of the exponent tells us the power on the base, and the denominator tells us the index of the root of the base. We can perform these operations on the base in either order. It’s generally easier to take the nth root of the base first and then raise that root to the power. For example, 93/2 can be written as (93 )1/2 or (91/2 )3 . Below, we simplify each expression:

  The calculation is easier to do, especially without a calculator, when we use the second form of the expression. We can check our work on a calculator if desired. We just have to make sure we use parentheses where needed. Example 2 illustrates how we do this.

  Example 2

  Evaluate the following expressions, and then check your work using a calculator.

  1.

      16¾

  2.

      

  3.

      

  4.

      (−27)⅔

  5.

      

  6.

      

  Solutions

  1.

    

  2.

    

  3.

    

  4.

    

  5.

    

  6.

    

  We can again check our solutions using a graphing calculator. The checks for problems 1–5 are show below. For problems 5 and 6, change a decimal answer to the fraction form by pressing the MATH button and selecting 1:Frac.

  Practice B

  Evaluate the following expressions, and then turn the page to check your work.

  7.

      

  8.

      (−32)⅖

  9.

      49−½

  10.

    

  11.

    216⅔

  12.

    81−¼

  C. Properties of Rational Exponents

  All of the properties of exponents that we learned for integer exponents also hold for Rational Exponents. Next we will apply those rules to simplify expressions involving Rational Exponents.

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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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  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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