Exponential Rules

4 Key excerpts on "Exponential Rules"

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  eBook - PDF

  Intermediate Algebra

  Connecting Concepts through Applications

  • Mark Clark, Cynthia Anfinson(Authors)
  • 2018(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. S E C T I O N 5 . 2 S o l v i n g E q u a t i o n s U s i n g E x p o n e n t R u l e s 489 Recap of the Rules for Exponents In Section 3.1, we studied many of the rules for exponents. In this section, we will use some of these rules to solve different types of equations. Solving Equations Using Exponent Rules LEARNING OBJECTIVES Solve power equations using exponent rules. Solve exponential equations by inspection. Identify exponential equations and power equations. 5.2 Rules for Exponents 1. x m # x n 5 x m 1 n 2. 1 xy 2 m 5 x m y m 3. x m x n 5 x m 2 n x ? 0 4. a x y b m 5 x m y m y ? 0 5. 1 x m 2 n 5 x mn 6. x 0 5 1 0 0 5 undefined 7. x 2n 5 1 x n x ? 0 DEFINITION Power Equation An equation of the form x n 5 a where a and n are any real numbers is called a power equation. Solving Power Equations We will use rational exponents to solve power equations. Raising both sides of an equation to the reciprocal exponent undoes exponents of variables we are trying to solve for. The square root property that we learned in Chapter 4 is an example of this process. Recall that the square root property required a plus/minus symbol to account for both possible solutions. The plus/minus symbol is necessary whenever we take an even root or raise both sides to a fractional exponent where the denominator is an even number. Odd power. Even power requires plus/minus symbol. x 5 5 32 x 4 5 81 1 x 5 2 1 5 5 1 322 1 5 1 x 4 2 1 4 5 6 1 812 1 4 x 5 2 x 5 63 Raising both sides of an equation to the reciprocal power finds only the real number solutions to the equation. In Chapter 8, we will study about other possible solutions to power equations.

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  eBook - PDF

  Prealgebra

  • Alan Tussy, Diane Koenig(Authors)
  • 2018(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  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. 10.1 • Multiplication Rules for Exponents 935 This example illustrates the following rule for exponents. Power Rule for Exponents To raise an exponential expression to a power, keep the base and multiply the exponents. For any number x and any natural numbers m and n, ( x m ) n 5 x m? n 5 x mn Read as “the quantity of x to the mth power raised to the n th power equals x to the mn th power.” THE LANGUAGE OF ALGEBRA An exponential expression raised to a power, such as (2 3 ) 7 , is also called a power of a power. Simplify: a. (2 3 ) 7 b. [(26) 2 ] 5 c. (z 8 ) 8 Strategy In each case, we want to write an equivalent expression using one base and one exponent. We will use the power rule for exponents to do this. WHY Each expression is a power of a power. Solution a. (2 3 ) 7 5 2 3?7 5 2 21 Keep the base, 2, and multiply the exponents. Since 2 21 is a very large number, we will leave the answer in this form. b. [(26) 2 ] 5 5 (26) 2?5 5 (26) 10 Keep the base, 26, and multiply the exponents. Since (26) 10 is a very large number, we will leave the answer in this form. c. (z 8 ) 8 5 z 8?8 5 z 64 Keep the base, z, and multiply the exponents. EXAMPLE 4 Simplify: a. (4 6 ) 5 b. (y 5 ) 2 Now Try Problems 49, 51, and 53 Self Check 4 Simplify: a. (x 2 x 5 ) 2 b. (z 2 ) 4 (z 3 ) 3 Strategy In each case, we want to write an equivalent expression using one base and one exponent. We will use the product and power rules for exponents to do this.

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  eBook - PDF

  Encyclopedia of Mathematics Education

  • Louise Grinstein, Sally I. Lipsey(Authors)
  • 2001(Publication Date)
  • Routledge

    (Publisher)

  Rudin, Walter. Principles of Mathematical Analysis. 3rd ed. New York: McGraw-Hill, 1976. ALEJANDRO ANDREOTTI ANN E. MOSKOL EXPONENTS, ARITHMETIC Rational numbers, written as superscripts, to indicate multiplication of identical quantities or ex-traction of roots. A positive integer exponent (or power) indicates multiplication of identical quanti-ties. For example, in 34 the exponent 4 indicates that four threes are multiplied together, namely, 34 = 3 X 3 X 3 X 3 = 81. The numeral 3 is called the base. If the exponent is a rational number, but not equivalent to a positive integer, then the meaning of the exponent is as follows: x° = 1, x~p = l/xp, x llp = pVx, for p a positive integer and x =£ 0 when it appears in the denomina-tor. For example, 5° = 1, 3-2 = l/(32) = 1/9, and 81/3 = /8 = 2. If n = plq> where p and q are both integers, and q ± 0, then xn = x p/q = ( xp) Vq = (. x Vq)p. For example, 8273 = 4. The idea of powers of numbers is an old idea. There is even a hint of it in the Egyptian Rhind pa-242 EXPONENTS, ARITHMETIC pyrus which dates from 1650 B.c. However, our modern superscript notation first appeared in Rene Descartes’ Discours de la Methode (1637). For some unexplained reason, even though Descartes would use x3 and x4, he never used x2, instead he used xx. The Pythagoreans (ca. 450 B.c.) classified numbers by shape. The square numbers were the ones that could be represented by arranging dots or squares in a square. Cubic numbers could be represented by building cubes. The Greeks had thought of x2 and x3 as areas and volumes of squares and cubes, whereas Descartes considered them as lengths of lines. Our reading x2 as “x-squared” and x3 as “x-cubed” is a legacy from the Pythagoreans. In 1676, Isaac New-ton introduced the idea of fractional and negative ex-ponents. The rules for multiplying and dividing quantities involving exponents can be found in Dio- phantus’ Arithmetica (mid-third century).

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  eBook - ePub

  Elementary Algebra

  • Toby Wagner(Author)
  • 2021(Publication Date)
  • Chemeketa Press

    (Publisher)

  The last two expressions from the previous example illustrate the importance of parentheses when applying exponents. Based on the order of operations, an exponent must be applied before multiplication takes place — unless the multiplication is inside of parentheses. In other words, we should only apply an exponent to the thing that appears immediately to the left of it.

  You may recall seeing this in previous math classes with expressions involving negative numbers. Remember that a negative sign signifies that something is being multiplied by −1. In the absence of parentheses, we must apply an exponent before a negative sign. Let’s see how this looks with the number −3:

  Now, let’s look at a couple of expressions involving the quotient rule and the zero-exponent rule.

  Example 8

  Simplify the following expressions.

  1.

    

  2.

    

  Solutions

  1.

    

  In this problem, we use the two-step process described before example 6.

  2.

    

  In this problem, we treat (x + 4) and (x – 1) as single entities and use the same approach we used in the first expression. We use the quotient rule to combine the (x + 4) and (x – 1) expressions.

  Now we’re ready to tackle the problem from the beginning of this section. This problem requires us to use all of the rules of exponents we’ve learned so far.

  Example 9

  Simplify:

  Solution

  We will start by multiplying these fractions together. This will require us to use the product rule for exponents. After that, as we work through the rest of the problem, the quotient rule and the zero-exponent rule will be used as well.

  In the previous example, the numerical coefficients gave us the fraction when multiplied. This fraction then had to be simplified. However, when multiplying fractions, we can use cross-canceling to reduce the numerical coefficients before combining the fractions. In the previous example, that step would look like this:

  By cross-canceling, we immediately arrive at the simplified fraction

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