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Prealgebra

Key Concepts

PrealgebraKey Concepts
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. 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
  17. Index

10.2 Use Multiplication Properties of Exponents

  • Exponential Notation On the left side, a raised to the m is shown. The m is labeled in blue as an exponent. The a is labeled in red as the base. On the right, it says a to the m means multiply m factors of a. Below this, it says a to the m equals a times a times a times a, with m factors written below in blue.

    This is read aa to the mthmth power.

  • Product Property of Exponents
    • If aa is a real number and m,nm,n are counting numbers, then
      am·an=am+nam·an=am+n
    • To multiply with like bases, add the exponents.
  • Power Property for Exponents
    • If aa is a real number and m,nm,n are counting numbers, then
      (am)n =amn(am)n =amn
  • Product to a Power Property for Exponents
    • If aa and bb are real numbers and mm is a whole number, then
      (ab)m=ambm(ab)m=ambm

10.3 Multiply Polynomials

  • Use the FOIL method for multiplying two binomials.
    Step 1. Multiply the First terms. .
    Step 2. Multiply the Outer terms.
    Step 3. Multiply the Inner terms.
    Step 4. Multiply the Last terms.
    Step 5. Combine like terms, when possible.
  • Multiplying Two Binomials: To multiply binomials, use the:
    • Distributive Property
    • FOIL Method
    • Vertical Method
  • Multiplying a Trinomial by a Binomial: To multiply a trinomial by a binomial, use the:
    • Distributive Property
    • Vertical Method

10.4 Divide Monomials

  • Equivalent Fractions Property
    • If a,b,ca,b,c are whole numbers where b0,c0,b0,c0, then
      ab=a·cb·canda·cb·c=abab=a·cb·canda·cb·c=ab
  • Zero Exponent
    • If aa is a non-zero number, then a0=1.a0=1.
    • Any nonzero number raised to the zero power is 1.1.
  • Quotient Property for Exponents
    • If aa is a real number, a0,a0, and m,nm,n are whole numbers, then
      aman=amn,m>nandaman=1anm,n>maman=amn,m>nandaman=1anm,n>m
  • Quotient to a Power Property for Exponents
    • If aa and bb are real numbers, b0,b0, and mm is a counting number, then
      (ab)m=ambm(ab)m=ambm
    • To raise a fraction to a power, raise the numerator and denominator to that power.

10.5 Integer Exponents and Scientific Notation

  • Summary of Exponent Properties
    • If a,ba,b are real numbers and m,nm,n are integers, then
      Product Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power Property(ab)m=ambmQuotient Propertyaman=amn,a0Zero Exponent Propertya0=1,a0Quotient to a Power Property(ab)m=ambm,b0Definition of Negative Exponentan=1anProduct Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power Property(ab)m=ambmQuotient Propertyaman=amn,a0Zero Exponent Propertya0=1,a0Quotient to a Power Property(ab)m=ambm,b0Definition of Negative Exponentan=1an
  • Convert from Decimal Notation to Scientific Notation: 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, nn, 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 10n10n.
      • If the original number is between 0 and 1, the power of 10 will be 10n10n.
    4. Step 4. Check.
  • Convert Scientific Notation to Decimal Form: To convert scientific notation to decimal form:
    1. Step 1. Determine the exponent, nn, on the factor 10.
    2. Step 2. Move the decimal nn places, adding zeros if needed.
      • If the exponent is positive, move the decimal point nn places to the right.
      • If the exponent is negative, move the decimal point |n||n| places to the left.
    3. Step 3. Check.

10.6 Introduction to Factoring Polynomials

  • Find the greatest common factor.
    1. Step 1. Factor each coefficient into primes. Write all variables with exponents in expanded form.
    2. Step 2. List all factors—matching common factors in a column. In each column, circle the common factors.
    3. Step 3. Bring down the common factors that all expressions share.
    4. Step 4. Multiply the factors.
  • Distributive Property
    • If aa, bb, cc are real numbers, then
      a(b+c)=ab+aca(b+c)=ab+ac and ab+ac=a(b+c)ab+ac=a(b+c)
  • Factor the greatest common factor from a polynomial.
    1. Step 1. Find the GCF of all the terms of the polynomial.
    2. Step 2. Rewrite each term as a product using the GCF.
    3. Step 3. Use the Distributive Property ‘in reverse’ to factor the expression.
    4. Step 4. Check by multiplying the factors.
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