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Prealgebra 2e

Key Concepts

Prealgebra 2eKey Concepts

Key Concepts

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 10-n10-n.
    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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