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7th Grade Pre-algebra Chapter 4 Notes 4.1 Factors and Monomials Vocabulary Factors: Two or more numbers that are multiplied to for a product Divisible: able to be divided without a remainder Monomial: a number, variable, or a product of numbers and/or variables. Ex. 5, 4x, 3xy Factors Find all the factors of 36 Find all the factors of 72 4.2 Powers and Exponents Vocabulary Base: the number that is multiplied Exponent: the ‘raised’ number, tells how many times the base is used as a factor Power: the number that can be expressed using an exponent is called a power 34 exponent base Standard form: a number expresses without exponents Expanded form: a number expressed using place values and exponents Remember Order of Operations Read it, Write it, factor it Read it Write it Factor it 2 to the first power 35 5∙5∙5∙5∙5 4 to the sixth power Evaluate the expressions 3 1. 2 2. y2 + 5 if y = -3 3. 3(x + y)4 if x = -2 and y = 1 4.3 Prime Factorization Vocabulary Prime Number: a whole number that has exactly two factors, 1 and itself Composite Number: a whole number that has more than two factors. ZERO and ONE ARE NEITHER PRIME NOR COMPOSITE Prime Factorization: when a composite number is expressed as the product of prime factors Factor tree: a way to find the prime factorization of a number using branches of the numbers factors Factor: to write a number as a product of its factors Prime or Composite? • 2 • 6 • 9 • 15 • 11 Prime Factorization • Use a factor tree to find the prime factorization of 280 You Try … Find the prime factorization of 392 Factoring Monomials Factor 8ab2 Factor -30x3y You Try… • Factor 64n3 • Factor -120r2st3 4.4 Greatest Common Factor (GCF) Vocabulary Venn Diagram: shows the relationship among sets of numbers or objects by using overlapping circles Greatest Common Factor: The greatest factor of two or more numbers Finding GCF Method 1: List out factors find the GCF of 12 and 20 Factors of 12: 1, 2, 3, 4, 6, 12 Factors of 20: 1, 2, 4, 5, 10, 20 GCF of 12 and 20 is 4 Find the GCF of 36 and 65 List the factors: 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 64: 1, 2, 4, 8, 16, 32, 64 GCF is : 4 Method 2: Using prime factorization Find the GCF of 30 and 24 Step 1: find prime factors using factor tree 30 24 6 5 6 4 2 3 2 2 3 2 2∙3∙5 2∙2∙2∙3 Step 2: List the common prime factors Factors of 24 Factors of 30 Shared Factors 2 2 2 3 2 3 5 2 3 Step 3: Find the product of the common factors Shared Factors are 2 and 3 2∙3 = 6 GCF of 30 and 24 is 6 Use Prime factorization to find the GCF of 252 and 126 252 126 126 2 63 2 63 2 21 3 9 7 7 3 3 3 7∙3∙3∙2 2∙2∙3∙3∙7 Common factors : 2 ∙ 3 ∙ 3 ∙ 7 GCF = 126 Finding GCF of Monomials Find the GCF of 16xy2 and 30xy 16xy2 30xy 4 4 x y y 6 5 x y 2 2 2 2 3 2 2∙2∙2∙2∙x∙y∙y 5∙3∙2∙x∙y Common Factors: 2 ∙ x ∙ y GCF is 2xy Finding GCF of Monomials Find the GCF of 12x and 40x2 12x 40x2 8 5 x x 3 4 x 4 2 2 2 Common Factors: 2 ∙ 2 ∙ x 2 2 GCF is 4x 2∙2∙3∙x 5∙2∙2∙2∙ x∙x Problem Solving There are 208 boys and 240 girls participating in a field day competition. What is the greatest number of teams that can be formed if each team has the same number of girls and each team has the same number of boys? HINT: FIND GCF of 208 and 240 Answer: 16 teams 4.5 Simplifying Algebraic Fractions Vocabulary Simplest form: when the numerator and denominator of a fraction have a GCF of 1 Algebraic Fraction: A fraction with variables in the numerator or denominator Simplifying Fractions Method 1: Divide the numerator and denominator by their GCF Simplify Simplifying Fractions Method 2: Factor the numerator and denominator and divide by all common factors. Simplify Simplify You Try… Simplify each fraction. Problem Solving There are 5280 feet in 1 mile. Eighty-eight feet is what part of 1 mile? 4.6 Multiplying and Dividing Monomials How this work.. 23 ∙ 24 = 2∙2∙2 ∙ 2 ∙ 2 ∙ 2 ∙ 2 = 27 53 ∙ 52 = 53+2 = 55 How this work.. 26 ÷ 23 = 2∙2∙2∙2∙2∙2 = 23 2∙2∙2 55 ÷ 52 = 55-2 = 53 You Try… Multiplying and Dividing Monomials Step 1: multiply or divide the coefficients (not common bases) Step 2: add or subtract the exponents of the common bases Find the product: 3n3 ∙ 2n4 = Find the quotient: 15x9 = 3x6 You try… 1. 3x5 ∙ 2x3 2. -6b3 ∙ 4b2 3. 18y6 ÷ 6y4 Two step problems Find each quotient or product 1. n3(n5) n2 2. k 3 m 2 k m 4.7 Negative Exponents Power Value Copy the table. 26 64 Describe the pattern of the powers in the right 25 32 column, then continue 24 16 the pattern by writing the next two powers in the 23 8 table. 22 4 Describe the pattern of the values in the second 21 2 column, then complete 20 the second column. 2-1 Negative Exponents • Method 1: x3 = x3-5 = x-2 x5 • Method 2: x3 = x∙x∙x = 1 x5 x∙x∙x∙x∙x x2 Therefore…. x-2 = 1 x2 • To write an expression using only positive exponents – Flip the position of the term • Move negative denominator values to numerator • Move negative numerator values to denominator – Keep the base the same – Change the negative sign to positive More examples Write each expression using positive exponents -3 5 = a-6 = (-6)-4 = Write each expression using a negative exponent other than -1 4.8 Scientific Notation Vocabulary Scientific Notation: a number expressed as a product of a factor and a power of 10. The factor must be greater than or equal to 1 and less than 10 Express Numbers in Standard Form • 3.78 x 106 Positive exponent: move decimal point to the right the same number of times of • 5.1 x 10-5 the exponent Negative Exponents: move the decimal point to the left the same number of times of the exponent Express Numbers in Scientific Notation • 60,000,000 • 0.0049 You Try… You Try…