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Intermediate Algebra

5.2 Properties of Exponents and Scientific Notation

Intermediate Algebra5.2 Properties of Exponents and Scientific Notation
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Use the Language of Algebra
    3. 1.2 Integers
    4. 1.3 Fractions
    5. 1.4 Decimals
    6. 1.5 Properties of Real Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations
    1. Introduction
    2. 2.1 Use a General Strategy to Solve Linear Equations
    3. 2.2 Use a Problem Solving Strategy
    4. 2.3 Solve a Formula for a Specific Variable
    5. 2.4 Solve Mixture and Uniform Motion Applications
    6. 2.5 Solve Linear Inequalities
    7. 2.6 Solve Compound Inequalities
    8. 2.7 Solve Absolute Value Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Graphs and Functions
    1. Introduction
    2. 3.1 Graph Linear Equations in Two Variables
    3. 3.2 Slope of a Line
    4. 3.3 Find the Equation of a Line
    5. 3.4 Graph Linear Inequalities in Two Variables
    6. 3.5 Relations and Functions
    7. 3.6 Graphs of Functions
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Systems of Linear Equations
    1. Introduction
    2. 4.1 Solve Systems of Linear Equations with Two Variables
    3. 4.2 Solve Applications with Systems of Equations
    4. 4.3 Solve Mixture Applications with Systems of Equations
    5. 4.4 Solve Systems of Equations with Three Variables
    6. 4.5 Solve Systems of Equations Using Matrices
    7. 4.6 Solve Systems of Equations Using Determinants
    8. 4.7 Graphing Systems of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Polynomials and Polynomial Functions
    1. Introduction
    2. 5.1 Add and Subtract Polynomials
    3. 5.2 Properties of Exponents and Scientific Notation
    4. 5.3 Multiply Polynomials
    5. 5.4 Dividing Polynomials
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Factoring
    1. Introduction to Factoring
    2. 6.1 Greatest Common Factor and Factor by Grouping
    3. 6.2 Factor Trinomials
    4. 6.3 Factor Special Products
    5. 6.4 General Strategy for Factoring Polynomials
    6. 6.5 Polynomial Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Rational Expressions and Functions
    1. Introduction
    2. 7.1 Multiply and Divide Rational Expressions
    3. 7.2 Add and Subtract Rational Expressions
    4. 7.3 Simplify Complex Rational Expressions
    5. 7.4 Solve Rational Equations
    6. 7.5 Solve Applications with Rational Equations
    7. 7.6 Solve Rational Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Roots and Radicals
    1. Introduction
    2. 8.1 Simplify Expressions with Roots
    3. 8.2 Simplify Radical Expressions
    4. 8.3 Simplify Rational Exponents
    5. 8.4 Add, Subtract, and Multiply Radical Expressions
    6. 8.5 Divide Radical Expressions
    7. 8.6 Solve Radical Equations
    8. 8.7 Use Radicals in Functions
    9. 8.8 Use the Complex Number System
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Quadratic Equations and Functions
    1. Introduction
    2. 9.1 Solve Quadratic Equations Using the Square Root Property
    3. 9.2 Solve Quadratic Equations by Completing the Square
    4. 9.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 9.4 Solve Quadratic Equations in Quadratic Form
    6. 9.5 Solve Applications of Quadratic Equations
    7. 9.6 Graph Quadratic Functions Using Properties
    8. 9.7 Graph Quadratic Functions Using Transformations
    9. 9.8 Solve Quadratic Inequalities
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Exponential and Logarithmic Functions
    1. Introduction
    2. 10.1 Finding Composite and Inverse Functions
    3. 10.2 Evaluate and Graph Exponential Functions
    4. 10.3 Evaluate and Graph Logarithmic Functions
    5. 10.4 Use the Properties of Logarithms
    6. 10.5 Solve Exponential and Logarithmic Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Conics
    1. Introduction
    2. 11.1 Distance and Midpoint Formulas; Circles
    3. 11.2 Parabolas
    4. 11.3 Ellipses
    5. 11.4 Hyperbolas
    6. 11.5 Solve Systems of Nonlinear Equations
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  13. 12 Sequences, Series and Binomial Theorem
    1. Introduction
    2. 12.1 Sequences
    3. 12.2 Arithmetic Sequences
    4. 12.3 Geometric Sequences and Series
    5. 12.4 Binomial Theorem
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  14. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
    12. Chapter 12
  15. Index

Learning Objectives

By the end of this section, you will be able to:
  • Simplify expressions using the properties for exponents
  • Use the definition of a negative exponent
  • Use scientific notation
Be Prepared 5.2

Before you get started, take this readiness quiz.

  1. Simplify: (−2)(−2)(−2).(−2)(−2)(−2).
    If you missed this problem, review Example 1.19.
  2. Simplify: 8x24y.8x24y.
    If you missed this problem, review Example 1.24.
  3. Name the decimal (−2.6)(4.21).(−2.6)(4.21).
    If you missed this problem, review Example 1.36.

Simplify Expressions Using the Properties for Exponents

Remember that an exponent indicates repeated multiplication of the same quantity. For example, in the expression am,am, the exponent m tells us how many times we use the base a as a factor.

First example: a raised to the power of m equals a times a times a times a and so on until you have multiplied m different factors of a together. Second example: the quantity negative 9 raised to the power of 5 equals negative 9 times negative 9 times negative 9 times negative 9 times negative 9, a total of 5 factors of negative 9.

Let’s review the vocabulary for expressions with exponents.

Exponential Notation

The figure shows the letter a in a normal font with the label base and the letter m in a superscript font with the label exponent. This means we multiply the number a with itself, m times.

This is read a to the mthmth power.

In the expression am,am, the exponent m tells us how many times we use the base a as a factor.

When we combine like terms by adding and subtracting, we need to have the same base with the same exponent. But when you multiply and divide, the exponents may be different, and sometimes the bases may be different, too.

First, we will look at an example that leads to the Product Property.

.
What does this mean? .
.

Notice that 5 is the sum of the exponents, 2 and 3. We see x2·x3x2·x3 is x2+3x2+3 or x5.x5.

The base stayed the same and we added the exponents. This leads to the Product Property for Exponents.

Product Property for Exponents

If a is a real number and m and n are integers, then

am·an=am+nam·an=am+n

To multiply with like bases, add the exponents.

Example 5.12

Simplify each expression: y5·y6y5·y6 2x·23x2x·23x 2a7·3a.2a7·3a.

Try It 5.23

Simplify each expression:

b9·b8b9·b8 42x·4x42x·4x 3p5·4p3p5·4p x6·x4·x8.x6·x4·x8.

Try It 5.24

Simplify each expression:

x12·x4x12·x4 10·10x10·10x 2z·6z72z·6z7 b5·b9·b5.b5·b9·b5.

Now we will look at an exponent property for division. As before, we’ll try to discover a property by looking at some examples.

Consider x5x2x5x2 and x2x3x2x3
What do they mean? x·x·x·x·xx·xx·x·x·x·xx·x x·xx·x·xx·xx·x·x
Use the Equivalent Fractions Property. x·x·x·x·xx·xx·x·x·x·xx·x x·x·1x·x·xx·x·1x·x·x
Simplify. x3x3 1x1x

Notice, in each case the bases were the same and we subtracted exponents. We see x5x2x5x2 is x52x52 or x3x3. We see x2x3x2x3 is or 1x.1x. When the larger exponent was in the numerator, we were left with factors in the numerator. When the larger exponent was in the denominator, we were left with factors in the denominator--notice the numerator of 1. When all the factors in the numerator have been removed, remember this is really dividing the factors to one, and so we need a 1 in the numerator. xx=1xx=1. This leads to the Quotient Property for Exponents.

Quotient Property for Exponents

If a is a real number, a0,a0, and m and n are integers, then

aman=amn,m>nandaman=1anm,n>maman=amn,m>nandaman=1anm,n>m

Example 5.13

Simplify each expression: x9x7x9x7 3103231032 b8b12b8b12 7375.7375.

Try It 5.25

Simplify each expression: x15x10x15x10 6146561465 x18x22x18x22 12151230.12151230.

Try It 5.26

Simplify each expression: y43y37y43y37 10151071015107 m7m15m7m15 98919.98919.

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like amam.amam. We know,xx=1,,xx=1, for any x(x0)x(x0) since any number divided by itself is 1.

The Quotient Property for Exponents shows us how to simplify amam.amam. when m>nm>n and when n<mn<m by subtracting exponents. What if m=n?m=n? We will simplifyamamamam in two ways to lead us to the definition of the Zero Exponent Property. In general, for a0:a0:

In the first way we write a to the power of m divided by a to the power of m as a to the power of the quantity m minus m. This is equal to a to the power of 0. In the second way we write a to the power of m divided by a to the power of m as a fraction with m factors of a in the numerator and a factors of m in the denominator. Simplifying this we can cross of all the factors and are left with the number 1. This shows that a to the power of 0 is equal to 1.

We see amamamam simplifies to a0a0 and to 1. So a0=1.a0=1. Any non-zero base raised to the power of zero equals 1.

Zero Exponent Property

If a is a non-zero number, then a0=1.a0=1.

If a is a non-zero number, then a to the power of zero equals 1.

Any non-zero number raised to the zero power is 1.

In this text, we assume any variable that we raise to the zero power is not zero.

Example 5.14

Simplify each expression: 9090 n0.n0.

Try It 5.27

Simplify each expression: 110110 q0.q0.

Try It 5.28

Simplify each expression: 230230 r0.r0.

Use the Definition of a Negative Exponent

We saw that the Quotient Property for Exponents has two forms depending on whether the exponent is larger in the numerator or the denominator. What if we just subtract exponents regardless of which is larger?

Let’s consider x2x5.x2x5. We subtract the exponent in the denominator from the exponent in the numerator. We see x2x5x2x5 is x25x25 or x−3.x−3.

We can also simplify x2x5x2x5 by dividing out common factors:

In the figure the expression x raised to the power of 2 divided by x raised to the power of 5 is written as a fraction with 2 factors of x in the numerator divided by 5 factors of x in the denominator. Two factors are crossed off in both the numerator and denominator. This only leaves 3 factors of x in the denominator. The simplified fraction is 1 divided by x to the power of 3.

This implies that x−3=1x3x−3=1x3 and it leads us to the definition of a negative exponent. If n is an integer and a0,a0, then an=1an.an=1an.

Let’s now look at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent.

1an Use the definition of a negative exponent,an=1an.11an Simplify the complex fraction.1·an1 Multiply.an 1an Use the definition of a negative exponent,an=1an.11an Simplify the complex fraction.1·an1 Multiply.an

This implies 1an=an1an=an and is another form of the definition of Properties of Negative Exponents.

Properties of Negative Exponents

If n is an integer and a0,a0, then an=1anan=1an or 1an=an.1an=an.

The negative exponent tells us we can rewrite the expression by taking the reciprocal of the base and then changing the sign of the exponent.

Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write the expression with only positive exponents.

For example, if after simplifying an expression we end up with the expression x−3,x−3, we will take one more step and write 1x3.1x3. The answer is considered to be in simplest form when it has only positive exponents.

Example 5.15

Simplify each expression: x−5x−5 10−310−3 1y−41y−4 13−2.13−2.

Try It 5.29

Simplify each expression: z−3z−3 10−710−7 1p−81p−8 14−3.14−3.

Try It 5.30

Simplify each expression: n−2n−2 10−410−4 1q−71q−7 12−4.12−4.

Suppose now we have a fraction raised to a negative exponent. Let’s use our definition of negative exponents to lead us to a new property.

(34)−2 Use the definition of a negative exponent,an=1an.1(34)2 Simplify the denominator.1916 Simplify the complex fraction.169 But we know that169is(43)2. This tells us that(34)−2=(43)2 (34)−2 Use the definition of a negative exponent,an=1an.1(34)2 Simplify the denominator.1916 Simplify the complex fraction.169 But we know that169is(43)2. This tells us that(34)−2=(43)2

To get from the original fraction raised to a negative exponent to the final result, we took the reciprocal of the base—the fraction—and changed the sign of the exponent.

This leads us to the Quotient to a Negative Power Property.

Quotient to a Negative Power Property

If a and b are real numbers, a0,b0a0,b0 and n is an integer, then

and(ab)n=(ba)nand(ab)n=(ba)n

Example 5.16

Simplify each expression: (57)−2(57)−2 (xy)−3.(xy)−3.

Try It 5.31

Simplify each expression: (23)−4(23)−4 (mn)−2.(mn)−2.

Try It 5.32

Simplify each expression: (35)−3(35)−3 (ab)−4.(ab)−4.

Now that we have negative exponents, we will use the Product Property with expressions that have negative exponents.

Example 5.17

Simplify each expression: z−5·z−3z−5·z−3 (m4n−3)(m−5n−2)(m4n−3)(m−5n−2) (2x−6y8)(−5x5y−3).(2x−6y8)(−5x5y−3).

Try It 5.33

Simplify each expression:

z−4·z−5z−4·z−5 (p6q−2)(p−9q−1)(p6q−2)(p−9q−1) (3u−5v7)(−4u4v−2).(3u−5v7)(−4u4v−2).

Try It 5.34

Simplify each expression:

c−8·c−7c−8·c−7 (r5s−3)(r−7s−5)(r5s−3)(r−7s−5) (−6c−6d4)(−5c−2d−1).(−6c−6d4)(−5c−2d−1).

Now let’s look at an exponential expression that contains a power raised to a power. See if you can discover a general property.

(x2)3 What does this mean?x2·x2·x2 (x2)3 What does this mean?x2·x2·x2

How many factors altogether? .
So we have .

Notice the 6 is the product of the exponents, 2 and 3. We see that (x2)3(x2)3 is x2·3x2·3 or x6.x6.

We multiplied the exponents. This leads to the Power Property for Exponents.

Power Property for Exponents

If a is a real number and m and n are integers, then

(am)n=am·n(am)n=am·n

To raise a power to a power, multiply the exponents.

Example 5.18

Simplify each expression: (y5)9(y5)9 (44)7(44)7 (y3)6(y5)4.(y3)6(y5)4.

Try It 5.35

Simplify each expression: (b7)5(b7)5 (54)3(54)3 (a4)5(a7)4.(a4)5(a7)4.

Try It 5.36

Simplify each expression: (z6)9(z6)9 (37)7(37)7 (q4)5(q3)3.(q4)5(q3)3.

We will now look at an expression containing a product that is raised to a power. Can you find this pattern?

(2x)3 What does this mean?2x·2x·2x We group the like factors together.2·2·2·x·x·x How many factors of 2 and ofx23·x3 (2x)3 What does this mean?2x·2x·2x We group the like factors together.2·2·2·x·x·x How many factors of 2 and ofx23·x3

Notice that each factor was raised to the power and (2x)3(2x)3 is 23·x3.23·x3.

The exponent applies to each of the factors! This leads to the Product to a Power Property for Exponents.

Product to a Power Property for Exponents

If a and b are real numbers and m is a whole number, then

(ab)m=ambm(ab)m=ambm

To raise a product to a power, raise each factor to that power.

Example 5.19

Simplify each expression: (−3mn)3(−3mn)3 (−4a2b)0(−4a2b)0 (6k3)−2(6k3)−2 (5x−3)2.(5x−3)2.

Try It 5.37

Simplify each expression: (2wx)5(2wx)5 (−11pq3)0(−11pq3)0 (2b3)−4(2b3)−4 (8a−4)2.(8a−4)2.

Try It 5.38

Simplify each expression: (−3y)3(−3y)3 (−8m2n3)0(−8m2n3)0 (−4x4)−2(−4x4)−2 (2c−4)3.(2c−4)3.

Now we will look at an example that will lead us to the Quotient to a Power Property.

(xy)3 This meansxy·xy·xy Multiply the fractions.x·x·xy·y·y Write with exponents.x3y3 (xy)3 This meansxy·xy·xy Multiply the fractions.x·x·xy·y·y Write with exponents.x3y3

Notice that the exponent applies to both the numerator and the denominator.

We see that (xy)3(xy)3 is x3y3.x3y3.

This leads to the Quotient to a Power Property for Exponents.

Quotient to a Power Property for Exponents

If aa and bb are real numbers, b0,b0, and mm is an integer, then

(ab)m=ambm(ab)m=ambm

To raise a fraction to a power, raise the numerator and denominator to that power.

Example 5.20

Simplify each expression:

(b3)4(b3)4 (kj)−3(kj)−3 (2xy2z)3(2xy2z)3 (4p−3q2)2.(4p−3q2)2.

Try It 5.39

Simplify each expression:

(p10)4(p10)4 (mn)−7(mn)−7 (3ab3c2)4(3ab3c2)4 (3x−2y3)3.(3x−2y3)3.

Try It 5.40

Simplify each expression:

(−2q)3(−2q)3 (wx)−4(wx)−4 (xy33z2)2(xy33z2)2 (2m−2n−2)3.(2m−2n−2)3.

We now have several properties for exponents. Let’s summarize them and then we’ll do some more examples that use more than one of the properties.

Summary of Exponent Properties

If a and b are real numbers, and m and n are integers, then

Property Description
Product Property am·an=am+nam·an=am+n
Power Property (am)n=am·n(am)n=am·n
Product to a Power (ab)m=ambm(ab)m=ambm
Quotient Property aman=amn,a0aman=amn,a0
Zero Exponent Property a0=1,a0a0=1,a0
Quotient to a Power Property (ab)m=ambm,b0(ab)m=ambm,b0
Properties of Negative Exponents an=1anan=1an and 1an=an1an=an
Quotient to a Negative Exponent (ab)n=(ba)n(ab)n=(ba)n

Example 5.21

Simplify each expression by applying several properties:

(3x2y)4(2xy2)3(3x2y)4(2xy2)3 (x3)4(x−2)5(x6)5(x3)4(x−2)5(x6)5 (2xy2x3y−2)2(12xy3x3y−1)−1.(2xy2x3y−2)2(12xy3x3y−1)−1.

Try It 5.41

Simplify each expression:

(c4d2)5(3cd5)4(c4d2)5(3cd5)4 (a−2)3(a2)4(a4)5(a−2)3(a2)4(a4)5 (3xy2x2y−3)2(9xy−3x3y2)−1.(3xy2x2y−3)2(9xy−3x3y2)−1.

Try It 5.42

Simplify each expression:

(a3b2)6(4ab3)4(a3b2)6(4ab3)4 (p−3)4(p5)3(p7)6(p−3)4(p5)3(p7)6 (4x3y2x2y−1)2(8xy−3x2y)−1.(4x3y2x2y−1)2(8xy−3x2y)−1.

Use Scientific Notation

Working with very large or very small numbers can be awkward. Since our number system is base ten we can use powers of ten to rewrite very large or very small numbers to make them easier to work with. Consider the numbers 4,000 and 0.004.

Using place value, we can rewrite the numbers 4,000 and 0.004. We know that 4,000 means 4×1,0004×1,000 and 0.004 means 4×11,000.4×11,000.

If we write the 1,000 as a power of ten in exponential form, we can rewrite these numbers in this way:

4,000 4×1,0004×1,000 4×1034×103
0.004 4×11,0004×11,000 4×11034×1103 4×10−34×10−3

When a number is written as a product of two numbers, where the first factor is a number greater than or equal to one but less than ten, and the second factor is a power of 10 written in exponential form, it is said to be in scientific notation.

Scientific Notation

A number is expressed in scientific notation when it is of the form

a×10nwhere1a<10andnis an integer.a×10nwhere1a<10andnis an integer.

It is customary in scientific notation to use as the ×× multiplication sign, even though we avoid using this sign elsewhere in algebra.

If we look at what happened to the decimal point, we can see a method to easily convert from decimal notation to scientific notation.

The figure shows two examples of converting from standard notation to scientific notation. In one example 4000 is converted to 4 times 10 to the power of 3. The decimal point in 4000 starts at the right and moves 3 places to the left to make the number 4. The 3 places moved make the exponent 3. In the other example, the number 0.004 is converted to 4 times 10 to the negative 3 power. The decimal point in 0.004 is moved 3 places to the right to make the number 4. The 3 places moved make the exponent negative 3.

In both cases, the decimal was moved 3 places to get the first factor between 1 and 10.

The power of 10 is positive when the number is larger than 1: 4,000=4×1034,000=4×103

The power of 10 is negative when the number is between 0 and 1: 0.004=4×10−30.004=4×10−3

How To

To convert a decimal to scientific notation.

  1. Step 1. Move the decimal point so that the first factor is greater than or equal to 1 but less than 10.
  2. Step 2. Count the number of decimal places, n, that the decimal point was moved.
  3. Step 3. Write the number as a product with a power of 10. If the original number is.
    • greater than 1, the power of 10 will be 10n.10n.
    • between 0 and 1, the power of 10 will be 10n.10n.
  4. Step 4. Check.

Example 5.22

Write in scientific notation: 37,000 0.0052.0.0052.

Try It 5.43

Write in scientific notation: 96,000 0.0078.

Try It 5.44

Write in scientific notation: 48,300 0.0129.

How can we convert from scientific notation to decimal form? Let’s look at two numbers written in scientific notation and see.

9.12×1049.12×10−4 9.12×10,0009.12×0.0001 91,2000.0009129.12×1049.12×10−4 9.12×10,0009.12×0.0001 91,2000.000912

If we look at the location of the decimal point, we can see an easy method to convert a number from scientific notation to decimal form.

The figure shows two examples of converting from scientific notation to standard notation. In one example 9.12 times 10 to the power of 4 is converted to 91200. The decimal point in 9.12 moves 4 places to the right to make the number 91200. In the other example, the number 9.12 times 10 to the power of -4 is converted to 0.000912. The decimal point in 9.12 is moved 4 places to the left to make the number 0.000912.

In both cases the decimal point moved 4 places. When the exponent was positive, the decimal moved to the right. When the exponent was negative, the decimal point moved to the left.

How To

Convert scientific notation to decimal form.

  1. Step 1. Determine the exponent, n, on the factor 10.
  2. Step 2. Move the decimal n places, adding zeros if needed.
    • If the exponent is positive, move the decimal point n places to the right.
    • If the exponent is negative, move the decimal point |n||n| places to the left.
  3. Step 3. Check.

Example 5.23

Convert to decimal form: 6.2×1036.2×103 −8.9×10−2.−8.9×10−2.

Try It 5.45

Convert to decimal form: 1.3×1031.3×103 −1.2×10−4.−1.2×10−4.

Try It 5.46

Convert to decimal form: −9.5×104−9.5×104 7.5×10−2.7.5×10−2.

When scientists perform calculations with very large or very small numbers, they use scientific notation. Scientific notation provides a way for the calculations to be done without writing a lot of zeros. We will see how the Properties of Exponents are used to multiply and divide numbers in scientific notation.

Example 5.24

Multiply or divide as indicated. Write answers in decimal form: (−4×105)(2×10−7)(−4×105)(2×10−7) 9×1033×10−2.9×1033×10−2.

Try It 5.47

Multiply or divide as indicated. Write answers in decimal form:

(−3×105)(2×10−8)(−3×105)(2×10−8) 8×1024×10−2.8×1024×10−2.

Try It 5.48

Multiply or divide as indicated. Write answers in decimal form:

; (−3×10−2)(3×10−1)(−3×10−2)(3×10−1) 8×1042×10−1.8×1042×10−1.

Access these online resources for additional instruction and practice with using multiplication properties of exponents.

Section 5.2 Exercises

Practice Makes Perfect

Simplify Expressions Using the Properties for Exponents

In the following exercises, simplify each expression using the properties for exponents.

81.

d3·d6d3·d6 45x·49x45x·49x 2y·4y32y·4y3 w·w2·w3w·w2·w3

82.

x4·x2x4·x2 89x·8389x·83 3z25·5z83z25·5z8 y·y3·y5y·y3·y5

83.

n19·n12n19·n12 3x·363x·36 7w5·8w7w5·8w a4·a3·a9a4·a3·a9

84.

q27·q15q27·q15 5x·54x5x·54x 9u41·7u539u41·7u53
c5·c11·c2c5·c11·c2

85.

mx·m3mx·m3

86.

ny·n2ny·n2

87.

ya·ybya·yb

88.

xp·xqxp·xq

89.

x18x3x18x3 5125351253 q18q36q18q36 102103102103

90.

y20y10y20y10 7167271672 t10t40t10t40 83858385

91.

p21p7p21p7 4164441644 bb9bb9 446446

92.

u24u3u24u3 9159591595 xx7xx7 1010310103

93.

200200 b0b0

94.

130130 k0k0

95.

270270 (270)(270)

96.

150150 (150)(150)

Use the Definition of a Negative Exponent

In the following exercises, simplify each expression.

97.

a−2a−2 10−310−3 1c−51c−5 13−213−2

98.

b−4b−4 10−210−2 1c−51c−5 15−215−2

99.

r−3r−3 10−510−5 1q−101q−10 110−3110−3

100.

s−8s−8 10−210−2 1t−91t−9 110−4110−4

101.

(58)−2(58)−2 (ba)−2(ba)−2

102.

(310)−2(310)−2 (2z)−3(2z)−3

103.

(49)−3(49)−3 (uv)−5(uv)−5

104.

(72)−3(72)−3 (3x)−3(3x)−3

105.

(−5)−2(−5)−2 5−25−2 (15)−2(15)−2 (15)−2(15)−2

106.

5−35−3 (15)−3(15)−3 (15)−3(15)−3 (−5)−3(−5)−3

107.

3·5−13·5−1 (3·5)−1(3·5)−1

108.

3·4−23·4−2 (3·4)−2(3·4)−2

In the following exercises, simplify each expression using the Product Property.

109.

b4b−8b4b−8 (w4x−5)(w−2x−4)(w4x−5)(w−2x−4) (−6c−3d9)(2c4d−5)(−6c−3d9)(2c4d−5)

110.

s3·s−7s3·s−7 (m3n−3)(m−5n−1)(m3n−3)(m−5n−1) (−2j−5k8)(7j2k−3)(−2j−5k8)(7j2k−3)

111.

a3·a−3a3·a−3 (uv−2)(u−5v−3)(uv−2)(u−5v−3) (−4r−2s−8)(9r4s3)(−4r−2s−8)(9r4s3)

112.

y5·y−5y5·y−5 (pq−4)(p−6q−3)(pq−4)(p−6q−3) (−5m4n6)(8m−5n−3)(−5m4n6)(8m−5n−3)

113.

p5·p−2·p−4p5·p−2·p−4

114.

x4·x−2·x−3x4·x−2·x−3

In the following exercises, simplify each expression using the Power Property.

115.

(m4)2(m4)2 (103)6(103)6 (x3)−4(x3)−4

116.

(b2)7(b2)7 (38)2(38)2 (k2)−5(k2)−5

117.

(y3)x(y3)x (5x)y(5x)y (q6)−8(q6)−8

118.

(x2)y(x2)y (7a)b(7a)b (a9)−10(a9)−10

In the following exercises, simplify each expression using the Product to a Power Property.

119.

(−3xy)2(−3xy)2 (6a)0(6a)0 (5x2)−2(5x2)−2 (−4y−3)2(−4y−3)2

120.

(−4ab)2(−4ab)2 (5x)0(5x)0 (4y3)−3(4y3)−3 (−7y−3)2(−7y−3)2

121.

(−5ab)3(−5ab)3 (−4pq)0(−4pq)0 (−6x3)−2(−6x3)−2 (3y−4)2(3y−4)2

122.

(−3xyz)4(−3xyz)4 (−7mn)0(−7mn)0 (−3x3)−2(−3x3)−2
(2y−5)2(2y−5)2

In the following exercises, simplify each expression using the Quotient to a Power Property.

123.

(p2)5(p2)5 (xy)−6(xy)−6 (2xy2z)3(2xy2z)3 (4p−3q2)2(4p−3q2)2

124.

(x3)4(x3)4 (ab)−5(ab)−5 (2xy2z)3(2xy2z)3 (x3yz4)2(x3yz4)2

125.

(a3b)4(a3b)4 (54m)−2(54m)−2 (3a-2b3c3)-2(3a-2b3c3)-2 (p-1q4r-4)2(p-1q4r-4)2

126.

(x2y)3(x2y)3 (103q)−4(103q)−4 (2x3y43z2)5(2x3y43z2)5 (5a3b-12c4)-3(5a3b-12c4)-3

In the following exercises, simplify each expression by applying several properties.

127.

(5t2)3(3t)2(5t2)3(3t)2 (t2)5(t−4)2(t3)7(t2)5(t−4)2(t3)7 (2xy2x3y−2)2(12xy3x3y−1)−1(2xy2x3y−2)2(12xy3x3y−1)−1

128.

(10k4)3(5k6)2(10k4)3(5k6)2 (q3)6(q−2)3(q4)8(q3)6(q−2)3(q4)8

129.

(m2n)2(2mn5)4(m2n)2(2mn5)4 (−2p−2)4(3p4)2(−6p3)2(−2p−2)4(3p4)2(−6p3)2

130.

(3pq4)2(6p6q)2(3pq4)2(6p6q)2 (−2k−3)2(6k2)4(9k4)2(−2k−3)2(6k2)4(9k4)2

Mixed Practice

In the following exercises, simplify each expression.

131.

7n−17n−1 (7n)−1(7n)−1 (−7n)−1(−7n)−1

132.

6r−16r−1 (6r)−1(6r)−1 (−6r)−1(−6r)−1

133.

(3p)−2(3p)−2 3p−23p−2 −3p−2−3p−2

134.

(2q)−4(2q)−4 2q−42q−4 −2q−4−2q−4

135.

(x2)4·(x3)2(x2)4·(x3)2

136.

(y4)3·(y5)2(y4)3·(y5)2

137.

(a2)6·(a3)8(a2)6·(a3)8

138.

(b7)5·(b2)6(b7)5·(b2)6

139.

(2m6)3(2m6)3

140.

(3y2)4(3y2)4

141.

(10x2y)3(10x2y)3

142.

(2mn4)5(2mn4)5

143.

(−2a3b2)4(−2a3b2)4

144.

(−10u2v4)3(−10u2v4)3

145.

(23x2y)3(23x2y)3

146.

(79pq4)2(79pq4)2

147.

(8a3)2(2a)4(8a3)2(2a)4

148.

(5r2)3(3r)2(5r2)3(3r)2

149.

(10p4)3(5p6)2(10p4)3(5p6)2

150.

(4x3)3(2x5)4(4x3)3(2x5)4

151.

(12x2y3)4(4x5y3)2(12x2y3)4(4x5y3)2

152.

(13m3n2)4(9m8n3)2(13m3n2)4(9m8n3)2

153.

(3m2n)2(2mn5)4(3m2n)2(2mn5)4

154.

(2pq4)3(5p6q)2(2pq4)3(5p6q)2

155.

(3x)2(5x)(3x)2(5x) (2y)3(6y)(2y)3(6y)

156.

(12y2)3(23y)2(12y2)3(23y)2 (12j2)5(25j3)2(12j2)5(25j3)2

157.

(2r−2)3(4−1r)2(2r−2)3(4−1r)2 (3x−3)3(3−1x5)4(3x−3)3(3−1x5)4

158.

(k−2k8k3)2(k−2k8k3)2

159.

(j−2j5j4)3(j−2j5j4)3

160.

(−4m−3)2(5m4)3(−10m6)3(−4m−3)2(5m4)3(−10m6)3

161.

(−10n−2)3(4n5)2(2n8)2(−10n−2)3(4n5)2(2n8)2

Use Scientific Notation

In the following exercises, write each number in scientific notation.

162.

57,000 0.026

163.

340,000 0.041

164.

8,750,000 0.00000871

165.

1,290,000 0.00000103

In the following exercises, convert each number to decimal form.

166.

5.2×1025.2×102 2.5×10−22.5×10−2

167.

−8.3×102−8.3×102 3.8×10−23.8×10−2

168.

7.5×1067.5×106 −4.13×10−5−4.13×10−5

169.

1.6×10101.6×1010 8.43×10−68.43×10−6

In the following exercises, multiply or divide as indicated. Write your answer in decimal form.

170.

(3×10−5)(3×109)(3×10−5)(3×109) 7×10−31×10−77×10−31×10−7

171.

(2×102)(1×10−4)(2×102)(1×10−4) 5×10−21×10−105×10−21×10−10

172.

(7.1×10−2)(2.4×10−4)(7.1×10−2)(2.4×10−4) 6×1043×10−26×1043×10−2

173.

(3.5×10−4)(1.6×10−2)(3.5×10−4)(1.6×10−2) 8×1064×10−18×1064×10−1

Writing Exercises

174.

Use the Product Property for Exponents to explain why x·x=x2.x·x=x2.

175.

Jennifer thinks the quotient a24a6a24a6 simplifies to a4.a4. What is wrong with her reasoning?

176.

Explain why 53=(−5)353=(−5)3 but 54(−5)4.54(−5)4.

177.

When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “simplify expressions using the properties for exponents.”, “use the definition of a negative exponent”, and “use scientific notation”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

After reviewing this checklist, what will you do to become confident for all goals?

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