Properties of Exponents

7 Key excerpts on "Properties of Exponents"

• 

  eBook - ePub

  Intermediate Algebra

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

    (Publisher)

  . Perform the division by canceling common factors.

  Notice that the exponent of the quotient is the difference between the exponents of the divisor and dividend. So the shortcut would look like this:

  In other words, when dividing exponential expressions with the same base, we write the result with the common base and subtract the exponents. This is the quotient property of exponents, which can be stated in general terms as

  For the time being, we must be aware of the condition m > n. Otherwise, the difference m − n could be zero or negative. We’ll explore those possibilities soon.

  There are three other basic Properties of Exponents. These rules can be demonstrated by expanding and simplifying each expression as we did with the product and quotient properties. The five basic Properties of Exponents are listed below.

  Five Properties of Exponents

  For any real numbers b and c and natural numbers m and n,

  Example 1

  Use the product and quotient properties to simplify the following expressions.

  1.

      b5 · b3

  2.

      (−2)5 · (−2)

  3.

      x2 · x5 · x3

  4.

      

  5.

      

  6.

      

  Solutions

  1.

    b5 · b3 = b3+5 = b8

  2.

    (−2)5 · (−2) = (−2)5 (−2)1 = (−2)6 = 64

  We can extend the product property of exponents to simplify a product of three factors having the same base. Simply add the exponents of all three factors:

  3.

    x2 · x5 · x3 = x2+5+3 = x10

  4.

    

  5.

    

  6.

    

  We can use all five Properties of Exponents in a variety of combinations to simplify more complex expressions. At first, it’s wise to proceed methodically using one property at a time. With practice, you may find that you’re able to combine some of the steps.

  Example 2

  Simplify the following expressions.

  1.

      (2 b3 c6 )4

  2.

      (5b c5 )(9 b4 c7 )

  3.

      

  4.

      

  Solutions

  1.

    (2b3 c6 )4

  2.

    (5bc5 )(9b4 c7 )

  3.

    

  4.

    

  Practice B

  Simplify the following expressions. Turn the page to check your work.

  1.

      t3 · t6 · t5

  2.

      (6a6 b3 )(4a9 b2 )

  3.

      

  4.

      (4x2 y7 )5

  5.

      

  C. Zero Exponents and Negative Exponents

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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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  GRE All the Quant

  Effective Strategies & Practice from 99th Percentile Instructors

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

    (Publisher)

  must equal 1.

  Exponent Rule 6:

  x0 = 1

  Any base raised to the power of zero is equal to 1.

  A Negative Exponent

  An exponent can be negative. While negative exponents have all the same properties as positive ones, they can be confusing to interpret. If you start from 22 and divide by the base, 2, you’ll move “down” one power from 22 to 21 .

  22 ÷ 2 = 22−1 = 21 = 2

  The below table illustrates what happens if you keep dividing by 2. The first column shows how to simplify an expression using plain math, while the second column shows how to simplify the exact same expression using exponent rules:

  Plain math:	Exponent form:	Therefore:
  21 ÷ 2 = 2 ÷ 2 = 1	21 ÷ 2 = 21−1 = 20	20 = 1
  	20 ÷ 2 = 20−1 = 2−1

  What happened there? When you drop below x0 to the negative exponent x–1 , the value doesn’t turn negative. Rather, the value turns into a fraction and the base ends up in the denominator of the fraction. This trend continues if you keep dividing by 2.

  When you see a negative exponent, first take the reciprocal of the base. In the above examples, the base is 2, so its reciprocal is .

  Next, put parentheses around the entire fraction and use the positive version of the exponent you started with. In the last example shown above, the starting exponent was −3, so the positive version is 3. As a result, .

  Exponent Rule 7:

  	For any base raised to a negative power, take the reciprocal of the base and raise it to the positive power.
  	For any base raised to the −1 power, take the reciprocal of the base and yo’re done.

  Here are three more examples of these rules:

  Check Your Skills

  1. If , what is y ?
  2. Simplify: (x−3 )(x−4 )

  Answers can be found on page 143.

  Special Bases

  The base of an exponent can be positive, negative, or zero. It could also be either an integer or a fraction. Chapter 2: Exponents introduced the basics of negative exponents. In this section, you’ll learn more about negative bases as well as other special bases.

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  AP Precalculus Premium, 2024: 3 Practice Tests + Comprehensive Review + Online Practice

  • Christina Pawlowski-Polanish(Author)
  • 2023(Publication Date)
  • Barrons Educational Services

    (Publisher)

  x).

  The power of a power law of exponents also leads to an interesting graphical transformation.

  Power Property and Change of Base

  The power property for exponents states that (b

  m

  )

  n

  = b

  (mn)

  . Graphically, this property implies that every horizontal dilation of an exponential function, f(x) = b

  (cx)

  , is equivalent to a change of the base of an exponential function, f(x) = (b

  c

  )

  x

  , where b

  c

  is a constant and c ≠ 0.

  Example

  Given the exponential function g(x) = 81

  x

  , state the factor of the horizontal dilation of g(x) if its parent function is f(x) = 3

  x

  .

  Solution

  Since 81 = 34 , then g(x) = 81

  x

  = (34 )

  x

  = 3

  4x

  . Therefore, the graph of g(x) is a horizontal shrink by a factor of of the graph of f(x).

  Example

  Show that the graph of g(x) = is a reflection over the y-axis of the graph of f(x) = 4

  x

  .

  Solution

  Using the negative exponent law, g(x) = = 4

  −x

  = f(−x). From the transformations section in this book, f(−x) reflects the graph of the original function, f(x), over the y-axis.

  Example

  Rewrite the function h(x) = in the form f(x) = ab

  x

  .

  Solution

  Using the exponent laws, .

  5.3 Exponential Function Context and Data Modeling

  Previous chapters discussed how to recognize if tables of data represented linear or quadratic functions. Exponential functions model growth patterns where successive output values over equal-length input value intervals are proportional. When the input values are whole numbers, exponential functions model situations of repeated multiplication of a constant to an initial value.

  Example

  Determine whether the functions in the tables below are linear, exponential, or neither. For those that are linear, find a linear function that models the data. For those that are exponential, find an exponential function that models the data.

  1.

  Table 5.1

  2.

  Table 5.2

  3.

  Table 5.3

  Solution

  For each table, the input values are of equal length, so calculate the average rate of change. If the average rate of change is constant, the function is linear. Also compute the ratio of consecutive outputs. If the ratio is constant, the function is exponential.

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  eBook - ePub

  The Parallel Curriculum in the Classroom, Book 2

  Units for Application Across the Content Areas, K-12

  • Carol Ann Tomlinson, Sandra N. Kaplan, Jeanne H. Purcell, Jann H. Leppien, Deborah E. Burns, Cindy A. Strickland(Authors)
  • 2005(Publication Date)
  • Corwin

    (Publisher)

  6

  The Power of Exponents

  A Middle or High School Math UnitIncorporating All Four Parallels Kristen Wogman Baron

  BACKGROUND FOR UNIT

  The topic of exponents and exponential functions is an important part of any Algebra I course. Students need to be able to multiply and divide exponential expressions, use scientific notation, and solve and graph simple exponential functions. This unit puts these initial concepts into a context to which students can relate and for which they can see a purpose.

  Students act as mathematicians in this unit as they discover the rules of exponents and the common characteristics of exponential function graphs. They also learn to use graphing calculator technology as a tool to aid them in computations. They act as scientists as they explore scientific notation of large and small numbers in the context of astronomy and microbiology. They act as financial planners as they learn to invest money for maximum profit.

  There are several options for assessment at the end of the unit. These options relate to the different fields of study we cover in the unit. Students choose an assessment based on their interest in a field area. This allows them to work in areas of interest to share their knowledge of the subject.

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  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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  Differentiating Instruction in Algebra 1

  Ready-to-Use Activities for All Students (Grades 7-10)

  • Kelli Jurek(Author)
  • 2021(Publication Date)
  • Routledge

    (Publisher)
• 4. Use the multiplication Properties of Exponents to simplify the following expressions.
1. 22 · 23 ────────
2. 54 · 55 ────────
3. a3 · a2 ────────
4. (43 )2 ────────
5. (d5 )3 ────────
   ─── I'm sure ─── I'm not sure
• 5. Use the division Properties of Exponents to simplify the following expressions.
1. m ─────────
2. v ────────
3. v ────────
4. v ────────
5. v ──────── ─── I'm sure ─── I'm not sure
6. 6. Use the zero and negative Properties of Exponents to simplify the following expressions.
   1. 20 ────────
   2. a0 · 24 ────────
   3. 3-3 ────────
   4. (4-3 )2 ────────
   5. (2d)-5 ────────
      ─── I'm sure ─── I'm not sure
7. 7. For the equation y = 450 (1.08 )4 identify the starting value, the growth factor, the time period, and the final value.
8. 1. Starting value:─────
   2. Growth factor:─────
   3. Time period: ─────
   4. Final value: ─────
   5. ─── I'm sure ─── I'm not sure
9. 8. Write an exponential function to model the following situation and then answer the question. Find the account balance after 3 years if you deposit $650 in an account that pays 4% annual interest compounded annually. ────────