Chapter 3: Factors, Roots, and Powers
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1 Chapter 3: Factors, Roots, and Powers Section 3.1 Chapter 3: Factors, Roots, and Powers Section 3.1: Factors and Multiples of Whole Numbers Terminology: Prime Numbers: Any natural number that has exactly two factors, 1 and the number itself. Prime Numbers Include: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, Composite Numbers: Any natural number that is not a prime number. Prime Factorization: The prime factorization of a number is the number written as a product of its prime factors. Determining the Prime Factors of a Whole Number Ex1. Write the prime factorization of each (a) 150 (b) 720 (c) 1250 (d)
2 Chapter 3: Factors, Roots, and Powers Section 3.1 (e) 2646 (f) 1440 Terminology: Greatest Common Factor: The greatest common factor of two or more numbers is the greatest factor the numbers have in common. Determining the Greatest Common Factor Ex1. Determine the greatest common factor of each set of numbers. (a) 138 and 198 (b) 126 and 144 (c) 175 and
3 Chapter 3: Factors, Roots, and Powers Section 3.1 Terminology: Least Common Multiple: The least common multiple of two or more numbers is the least number that is divisible by each number. Determining the Least Common Multiple Determine the least common multiple of each set of numbers. (a) 30 and 45 (b) 18 and 40 52
4 Chapter 3: Factors, Roots, and Powers Section 3.1 (c) 18, 20, and 30 (d) 28, 42, and 63 Practice Problems 5,7,8,9,10,11,15 pg
5 Chapter 3: Factors, Roots, and Powers Section 3.2 Section 3.2: Perfect Squares, Perfect Cubes, and Roots Terminology: Perfect Squares: A perfect square is a number that is produced by multiplying a number by itself (ie. squaring it). Perfect Squares Include: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, Perfect Cubes: A perfect cube is a number that is produced by multiplying a number by itself three times (ie. cubing it). Perfect Cubes Include: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, Radical: Consists of a root symbol, an index, and a radicand. It can be rational (for example 4) or irrational (for example 2). n The expression x is a radical where n is the index and x is the radicand. Square Root: A radical with an index of 2 (a square root often doesn t have a visible index, this is the only radical for which this is true). Cube Root: A radical with an index of 3. Determining the Square Root of a Whole Number Ex. Determine the value of each. (a) 1296 (b) 1764 NOTE: When dealing with square roots you must make two equal groups with your prime factorization. The product of the factors in each group is the value of the square root. 54
6 Chapter 3: Factors, Roots, and Powers Section 3.2 Determining the Cube Root of a Whole Number Ex. Determine the value of each. 3 (a) (b) 2744 NOTE: When dealing with cube roots you must make three equal groups with your prime factorization. The product of the factors in each group is the value of the square root. Using Roots to Solve a Problem Ex1. A cube has a volume of 4913 in 3. What is the surface area of the cube? Ex2. A cube has volume ft 3. What is the surface area of the cube? Practice Problems 4,5,7,8 pg
7 Chapter 3: Factors, Roots, and Powers Section 3.3 Section 3.3: Mixed and Entire Radicals Terminology: Mixed Radical Form: n 3 4 A radical of the form a x such as 4 5, 2 18, 3 2. A mixed radical is often called a simplified radical. Entire Radical Form: n 3 4 A radical of the form x such as 80, 144, 162. A entire radical cannot have a coefficient. Perfect Cubes Include: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000, Expressing Radicals in Simplified Radical Form (Mixed Radical) Determine the exact simplified radical form of each. (a) 72 (b) 26 (c) 500 (d)
8 Chapter 3: Factors, Roots, and Powers Section 3.3 (e) 245 (f) (f) 40 3 (g) 32 3 (h) (i) (j) (k) 32 57
9 Chapter 3: Factors, Roots, and Powers Section 3.3 Writing Mixed Radicals as Entire Radicals Ex. Express each mixed radical as an entire radical. (a) (b) (c) 2 2 (d) (e) (f) 2 3 Practice Questions 4,5,10,11,12,17,18,21 pg
10 Chapter 3: Factors, Roots, and Powers Section 3.5 Section 3.4: The Real Number System Terminology: Natural Numbers (N): All counting numbers starting with 1. Include: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, Whole Numbers (W): All Natural Numbers as well as Zero. Includes: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, Integers (I): All positive and negative Natural Numbers and Zero. Includes:, -3, -2, -1, 0, 1, 2, 3, Rational Numbers (Q): Any number that can be written in the form of a fraction. This includes repeating and terminating decimals. Irrational Numbers (Q ): Any number that cannot be written in the form of a fraction. This includes any nonrepeating, non-terminating decimal. Real Numbers (R): All natural, whole, integer, rational, or irrational number. The real number system can be demonstrated on a diagram as shown below: Real Number System (R) N W I Q Q Each of the number systems exists as a subsystem of the one outside it. ie. Any natural number is also a whole number, an integer, a rational number and also a real number. However any Integer is also a rational and real number. Irrational numbers are only subset of the real numbers. 59
11 Chapter 3: Factors, Roots, and Powers Section 3.5 Classifying Numbers as Subsystems of the Real Number System Classify each according to its subsets of the real number system. Display your answers on a diagram. 2, π, 7, 1 4, 9, 36, 0, 5 9 Practice Problems 3,4,7 pg
12 Chapter 3: Factors, Roots, and Powers Section 3.5 Section 3.5: The Laws of Exponents The Laws of Exponents: There are several laws that we use when we deal with the use of exponents in expressions and equations. Law 1: Product of Powers When we multiply two powers with the same base, we must add the values of the exponents. Ex: Write each as a single power: a m a n = a m+n (a) (c) ( 1 3 )5 ( 1 3 )9 (b) (d) ( 7) 2 ( 7) 4 ( 7) 7 Law 2: Quotient of Powers When we divide two powers with the same base, we must subtract the values of the exponents. Ex: Write each as a single power: a m a n = am n or a m a n = a m n (a) (c) (b) ( 11) 8 ( 11) 2 (d)
13 Chapter 3: Factors, Roots, and Powers Section 3.5 Law 3: Power of a Power When an exponent is applied to a power, we must multiply the powers of the two exponents. Ex: Write each as a single power: (a m ) n m n = a (a) (6 2 ) 4 (c) [( )4 ] (b) [( 11) 8 ] 5 (d) [ ]3 Law 4: Power of a Product When an exponent is applied to the product of two numbers, we use the distributive property and apply that exponent to each of the factors. Ex: Write each as a single power: (a) ( ) 4 (a b) m = a m b m (b) ( ) 3 62
14 Chapter 3: Factors, Roots, and Powers Section 3.5 Law 5: Power of a Quotient When an exponent is applied to the quotient of two numbers, we use the distributive property and apply that exponent to both the numerator and denominator (divisor and dividend). Ex: Write each as a single power: ( a b ) m = am b m (a) ( )2 (b) ( )5 Law 6: Power of Zero When a base has an exponent of zero, the value of the power is one. Ex: Write each as a single power: a 0 = 1 (a) 5 0 (b) ( 7) 0 (c) ( 1 5 )0 (d) ( ) 0 Law 7: Power of One When a base does not have an exponent, it actually has an exponent of one. a = a 1 63
15 Chapter 3: Factors, Roots, and Powers Section 3.5 Simplifying Exponential Expressions Involving Variables Sometimes, in exponential expressions, we encounter bases that are variables in addition to those that are constants like we have explored thus far. We deal with variable bases in the same way that we would deal with numerical bases. We just apply the laws of exponents in the same fashion as before, simplifying all bases that share the same variable. Ex. Simplify each expression using the laws of exponents. (a) a 2 b 3 a 5 b 2 (d) m 4 n 7 n 5 m 3 (b) (x 3 y 5 ) 3 (x 2 y) 5 (e) (a 4 b 2 ) 2 (ab 3 ) 4 (c) x5 y 4 z 8 x 2 y 3 z 5 (f) x3 y 5 z 7 x 3 y 2 z 4 64
16 Chapter 3: Factors, Roots, and Powers Section 3.5 (g) ( r5 s 2 t 4 r 2 st 3 )2 ( r2 s 5 t 3 r 2 s 2 t )3 (i) ( a2 b 3 c 5 abc 3 )3 ( a4 b 4 c 2 a 2 b )2 (h) ( 5x5 y 3 2xy 2 )3 (j) ( 7x5 y 3 3x 2 y )2 NOTE: If the word evaluate is used in a question a numerical answer is required. 65
17 Chapter 3: Factors, Roots, and Powers Section 3.5 Section 3.6: Negative Exponents and Reciprocals Complete the following table: Power Value What do you notice about the values of each power when comparing its value for the negative and corresponding positive exponent? 66
18 Chapter 3: Factors, Roots, and Powers Section 3.5 Law 8: Negative Exponents When a power has negative exponent, it can be rewritten as the reciprocal of the base with a positive exponent. Ex: Evaluate: a m = 1 a m or (a b ) m = ( b a ) m (a) 3 2 (d) 7 2 (b) ( 3 4 ) 3 (e) ( 10 3 ) 3 (c) (f) ( 1.5) 3 NOTE: When simplifying an exponential expression, the final answer cannot contain a negative exponent!!!! 67
19 Chapter 3: Factors, Roots, and Powers Section 3.6 Negative Exponents in Exponential Expressions with Variables Simplify each expression: (a) a3 b 2 c 4 a 2 b 4 c 3 (b) x5 y 2 z 5 x 6 y 5 z (c) x 3 y 3 x 2 y 2 (d) j 4 k 2 j 2 k 9 (e) a 2 b 5 c 2 a 6 b 2 c 3 (f) x2 y 3 z 5 x 2 y 4 z 2 66
20 Chapter 3: Factors, Roots, and Powers Section 3.6 (g) (3x 2 y 3 ) 2 (6xy 2 ) 2 (h) (4ab 3 ) 2 (2a 2 b 3 ) 3 (i) ( x5 y 2 z 3 x 2 y 5 z 2 )2 ( x2 y 2 z x 3 y 1 z 3) 1 (j) ( j3 k 4 l 7 j 5 k 2 l 3) 2 ( j 1 kl 5 j 5 k 2 l 3)3 67
21 Chapter 3: Factors, Roots, and Powers Section 3.6 ERROR ANALYSIS Identify any errors and write the correct solutions ( x 4 y 2 2 ) = (10x 6 y 6 ) 2 10x 2y 4 = (10) 2 x 12 y 12 = 20x 12 y 12 = 20x12 y 12 ERRORS: CORRECT SOLUTION: 68
22 Chapter 3: Factors, Roots, and Powers Section 3.7 Section 3.6: Rational/Fractional Exponents Law 9: Rational Exponents When a power has a rational exponent, its numerator represents the applied the exponent and the denominator represents the index of the applied radical(root). a m n n = a m or a m n n = ( a ) m Writing an Expression in Radical and Exponential Form A expression may be written in one of two ways when considering rational exponents. It may be given in exponential form or radical form. You may be asked to convert between these forms. Ex. Express each in its equivalent radical form: (a) (b) (c) x 3 4 (d) (e) (d) z 4 11 Ex. Express each as a single power 3 (a) 5 4 (b)( 7) 5 6 (c) x 3 4 (e) ( 9) 5 3 (e) 3 9 (d) ( x) 5 68
23 Chapter 3: Factors, Roots, and Powers Section 3.7 Evaluating Powers with Rational Exponents Evaluate each power. (Hint: Recommend that you apply the radical first then the exponent) (a) (b) ( 64) 2 3 (c) (0.0016) 3 4 (d) ( 32) 0.4 (e) ( 4 9 ) 3 2 (f) (g) ( 343) 2 3 (h) (i) (0.0625) 0.75 (j) ( 27 )
24 Chapter 3: Factors, Roots, and Powers Section (k) ( ) 2 1 (l) ( ) 4 Simplifying and Evaluating Rational Expressions Simplify to a single power and Evaluate (a) (b) (c) [( ) 4 ] [( )2 ] (d) [( )2 ] [( )4 ] (f) ( ) (1.4) 2 (1.5) 5 (e) (1.4)3 (1.4) 4 (g) ( ) 6 (h)
25 Chapter 3: Factors, Roots, and Powers Section 3.7 Rational Exponents in Exponential Expressions with Variables Simplify each expression: (a) (x 3 2y 2 ) (x 1 2y 1 ) (b) (x 3 y 1 2) (x 5 y 3 2) (c) (8a 3 b 6 ) 2 3 (d) (25a 4 b 2 ) (e) 4a 2 b3 2a 2 1 b3 5 (f) 12x 5 y2 1 3x2y
26 Chapter 3: Factors, Roots, and Powers Section 3.7 (g) ( 100a 25a 5 b 1 2 ) 1 2 (h) ( 50x2 y 4 1 ) 2 2x 4 y 7 Error Analysis Identify any error(s) in each solution and write the correct solution ERRORS: 1 (r 1 2 s 3 2 2) (r 1 4 s 1 2) 1 = r 1 s 1 r 5 4 s 1 2 = r s = r 1 4 s 3 2 = 1 r 1 4 s 3 2 CORRECT SOLUTION: Practice Problems 4,8,10,11,16,17,19,21 pg
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