Section 3.1 Factors and Multiples of Whole Numbers:

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1 Chapter Notes Math 0 Chapter : Factors and Products: Skill Builder: Some Divisibility Rules We can use rules to find out if a number is a factor of another. To find out if, 5, or 0 is a factor look at the last digits in the number is a factor when the last digit is even 5 is a factor of any number that ends in 0 or 5 0 is a factor when the last digit is 0 To find out if is a factor add the digits in the number to see if it is a factor of the sum Section. Factors and Multiples of Whole Numbers: We can factor in different ways: A prime number has only two factors: itself and The Prime factorization of is: The prime factorization of a number is the product of its prime factors Example : Write the prime factorization for each of the following (A) 50 (B) 7 (C)

2 Chapter Notes Math 0 The Greatest Common Factor (GCF) of two numbers is the greatest factor the numbers have in common: Example : Find the GCF between (A) 9 and 5 (B) and 8 Note: Prime factorization can be used to find the GCF Use the common primes and the lowest power (C) and 6 (D) and 0 Example : length of Two ribbons are cm and cm long. All pieces from both ribbons are to be cut into equal pieces. What is the greatest possible each piece?

3 Chapter Notes Math 0 The multiples of a number are found by multiplying it by a whole number. For example the multiples of are: The Least Common Multiple (LCM), is the lowest number that is divisible by both given numbers. Example : Find the LCM between (A) and 6 (B) and 0 Note: Prime factorization can be used to find the LCM Use all the primes and the highest power (C) 0 and 7 Example 5: Hamburger patties come in packages of 8. Buns come in packages of 6. What is the least number of hamburgers that can be made with no patties or buns left over? Questions page 0 #a,ae,5bce,6af, 8abe, 0abd

4 Chapter Notes Math 0 Solving Problems involving GCF and LCM Example 6: (A) What is the side length of the smallest square that could be tiled with rectangles that measure 6cm by 0cm? Assume the rectangles cannot be cut. 6cm 0cm (B) What is the side length of the largest square that could be used to tile a rectangle that measures 6cm by 0cm? Assume the rectangles cannot be cut. 0cm 6cm Questions page 0 #9a,a,,6ace, 7,9a

5 Chapter Notes Math 0 5. Perfect Squares, Perfect Cubes & their Roots: A Perfect Square is the square of a whole number. For example 6 and 00 are perfect squares. Since We say, is the square root of 6 We write, A Perfect Cube is the cube of a whole number. For example 8 and 5 are perfect cubes. Since We say, is the cube root of 8 We write, Terminology: A radical is an expression that has a square root, cube root, etc. Index Radicand Note: For square root the index is, it is something that is known and not written in. Example : Determine the square root (A) (B) (C) Example : Determine the cube root (A) (B) (C)

6 Chapter Notes Math 0 6 Using roots to solve a problem Example : (A) The volume of a cube is 5 cubic feet. What is the length of each side of the cube? (B) The volume of a rectangular prism is 55 m. The prism has a square cross section and a height of 5m. What is the length and width of the prism? Questions page 6-7 #, 5, 7, 8,, 7

7 Chapter Notes Math 0 7 Chapter : Roots and Powers: Section. Irrational Numbers: Set of Real Numbers These are rational numbers 0.5 These are irrational numbers Rational Numbers have exact answers. They can be written as a whole number, integer, fraction, repeating decimal or a terminating decimal. Irrational Numbers, the decimal representation does not terminate or repeat There are 6 number sets: Symbol N Natural = {,,,,...} the counting numbers greater than zero W I Q Whole = {0,,,,...} the counting numbers including zero Integers = {...,-,-,-,0,,,,...} all positive and negative numbers (no decimal or fractions) Rationals can written as a fraction or a decimal that ends (terminates) or repeats. Ex. Irrationals are numbers that in decimal form do not repeat or do not end. Ex.

8 Chapter Notes Math 0 8 Example : Classify (identify) which set(s) each number belongs to. (A) - (B) (C) (D) Ordering Numbers from Least to Greatest. Example : Rewrite from least to greatest Questions page #-6,, Section. Mixed and Entire Radicals: Examples of Radicals: Radicals have a Root symbol: Index Note: For square root the index is, it is something that is known and not written in. Radicand Entire Radical: All radicals under the root sign ex. Mixed Radical: has number and root part, ex.

9 Chapter Notes Math 0 9 Multiplication Property: Simplifying Square Roots: Example : This can also be broken down using the multiplication property. 6 has two perfect square factors Note: Not all radicals are perfect square numbers. To simplify non-perfect square radicands, find the highest perfect square factor of the radicand. {, 9, 6, 5, 6, 9, 6, 8, 00,,, 69, } Example : Simplify (A) (B) (C) Example : (A) Write as an entire radical (B) Questions page 8 #, 5bdfg, 0

10 Chapter Notes Math 0 0 Simplifying Cube Roots: Example : Simplify (A) (B) Perfect Cubes: {8, 7, 6, 5, } (C) (D) Example : (A) Write as an entire radical (B) Simplifying Radicals: Questions page 8 #, acegi

11 Chapter Notes Math 0 Square Root: Cube Root: (A) (B) th Root: (A) (B) 5 th Root: (A) (B) Questions page 8-9 #,5,7ab, 8ac,#0 Review: page 9 #a, a, a, 6a, 7a, 0 page #,, 7b, 9,

12 Chapter Notes Math 0 Section. Fractional Exponents and Radicals: Using your calculator, find (A) (B) Raising a number to the exponent is the same as finding square root. Raising a number to the exponent is the same as finding cube root. Example, rewrite as a radical (A) = (B) = Example, rewrite as a fractional power (A) (B) = Powers with Rational Exponents: or I index b radicand E exponent Example, rewrite as a radical (A) = (B) = Example, rewrite as a fractional power (A) (B) =

13 Chapter Notes Math 0 Note: handling fractions Apply power to the numerator and denominator Example: Evaluate (A) (B) (C) Questions page 7 #bce,5,6ad,7,8,0ae,ac NOTE: Power of zero: Converting a decimal to a fraction: Example: (A) 0. (B).5 Be able to write out the steps when evaluating: Example: (A) (B) Questions page 7-8 #ab,0cf,d,,6a,7, 8,9

14 Chapter Notes Math 0 Section.5 Negative Exponents and Reciprocals: Making a connection to negative powers. Complete Complete Write as a fraction A) = - = Write as fraction with base B) = - = C) = - = Complete A) = rewrite as with base B) = C) = NOTE: Switching the term from top to bottom, and vice versa, changes the sign on the power Ex. (A) (B)

15 Chapter Notes Math 0 5 Powers with Negative Exponents: or NOTE: ) Only the term with the negative power moves ) a negative power is not a negative number verses Example: Rewrite each as a positive exponent and then evaluate. (A) (B) (C) (D) To make outside power positive, Take reciprocal of inside fraction

16 Chapter Notes Math 0 6 (E) (F) Example: Given that, what is? Questions page #5,6,7,8 Negative Fractional Exponents: Examples: (A) (B)

17 Chapter Notes Math 0 7 (C) (D) (E) (F) Questions page - #9,,,5

18 Chapter Notes Math 0 8 Section.6 Applying the Exponent Laws : (I) Multiplication Property Evaluate: Multiplication Property: a m a n = (II) Division Property Division Property: Evaluate: (III)Power of a power Property Evaluate: ( ) (IV)Power of Product Evaluate: ( ) Power Property: (a m ) n = Power of Product: (ab) m = (V) Power of Quotient Power of Quotient: Evaluate: (VI) Evaluate: Zero Exponent Zero Exponent: x 0 = NOTE: Simplest form: Has NO negative power Has a decimal or a fraction, not both

19 Chapter Notes Math 0 9 Example : Simplify by expressing as a single power. (A) (B) (C) (D) Questions page - #ab,bcd,5b,6bc,7bc

20 Chapter Notes Math 0 0 Example : Simplify (a) (b) Example : Simplifying Algebraic expressions with Rational Exponents: (A) (B)

21 Chapter Notes Math 0 (c) (d) Questions page - #8cdegh,9cefgh,0cdeg,,,6

22 Chapter Notes Math 0 Unit Review: Powers and Roots Sections.-.,.-.6 Section. : Write the prime factorization of and 600 : Find the GCF of each pair of numbers: A) and 70 B) 6 and 8 : Find the LCM of each pair of numbers: A) and 0 B) 6 and 8 : Hamburger patties come in packages of 8. Buns come in packages of 6. What is the least number of hamburgers that can be made with no patties or buns left over? Section. 5: Use prime factorization to find each square root A) B) C) 6: Use prime factorization to find each cube root. A) B) C) Section. 7: Tell whether each number is rational or irrational. A) 6 B) C) 8 D) 8 8: Find each square root. Identify any that are irrational. A) B) 5 C) 9 D) 00 9: Find each cube root. Identify any that are irrational. Section. A) 6 B) 6 C) D) 00 0: Simplify each radical. A) 0 B) 75 C) D) 08 E) 0 F) 6 G) 89 H) 576

23 Chapter Notes Math 0 : Write each mixed radical as an entire radical. A) B) C) D) 5 Section. : Write each power as a radical. A) 8 B) C) : Write each radical as a power. A) 5 B) C) 6 : Evaluate each power. A) 00 B) 5 C) 8 5: Find the value of each power and radical to decimal places. A) B) 0 C) 50 6: Write each power as a radical in two ways. A) 5 B) C) 9 7: Write each radical as a power with a fractional exponent. A) 57 B) 5 8: Evaluate each power. Section.5 A) 5 B) 6 C) 6 9: Evaluate each power. C) 6 A) 6 B) 5 C) 5 D) 0: Write each power as a fraction with a radical denominator. 5 E) F) 0 A) B) C) 8

24 Chapter Notes Math 0 : Write each power with a positive exponent. A) B) C) 5 6 : Evaluate each power. A) 00 B) 5 C) 6 D) 00 E) 5 F) 6 5 G) Section H) 7 I). 9 : Write as a power with a positive exponent. (use a calculator) A) B) C) D) 8 8 : Evaluate. 7 A) B) 5 5 C) 5: Write as a power with a positive exponent. A) 5 5 B) C) D) : Simplify. Write an expression with positive exponents where necessary. A) y y y B) x x C) 5 b 0 7a D) b a E) x x F) 6 a a

25 Chapter Notes Math 0 5 ANSWERS Section.. = x ; 600 = x x 5. a. = x ; 70 = x 5 x 7; GCF = b. 6 = x ; 8 = x ; GCF = x =. a. = x ; 0 = x x 5; LCM = x x 5 = 60 b. 6 = ; 8 = x ; LCM = x = 6 x 9 =. 8 = ; 6 = x ; LCM = x = ; hamburgers need to be made so that no patties or buns are left over. Section.: 5. a. 5 b. c a. 9 b. 5 c. Section.: 7. a. Rational b. Irrational c. Rational d. Rational 8. a. b irrational c. d a. -6 b. c. d Irrational Section.: 0. a. 8 5 b. 7 5 c. 6 d. 6 e. 5 f. 6 g. 7 h. 9. a. 99 b. 5 c. 08 d. 0. a. 8 b. c.. a. 5 b. c. 6. a. 0 b. 5 c. 5. a..69 b.. c a. 7. a b. c. 9 9 b c a. 5 b. 6 c. 9. a) b) c) d) 6 e) 7 f) 0 0. a) b) c).a) b) c).a) b) c) d) e) f) g) h). a) 5 b) 5 c) d) 8. a) 9 b) c) 5.a) b) c) d) 6. a) y 5 b) c) d) e) f)