Prepared by Sa diyya Hendrickson. Package Summary

Transcription

1 Introduction Prepared by Sa diyya Hendrickson Name: Date: Figure 1: c Cengage Learning Package Summary Definition and Properties of Exponents Understanding Properties (Frayer Models) Discovering Zero and Negative Exponents Working with Fractions in Disguise Let s Play! (Exercises) Notes and Resources High School Math Prep 1 of 10 c Sa diyya Hendrickson

2 Some Properties Recall the following definition of a positive exponent: 1. The Exponential Form b n : For some number b and some positive integer n, b n simply means to multiply b exactly n times (i.e. repeated multiplication). Using algebra, we have: b n = b b b b }{{} n times The left side is the simplified form and the right side is the expanded form. Here, the base b has been raised to the exponent/power of n. e.g. ( 2) 3 = ( 2)( 2)( 2) = (2 2 2) = 8 Let a and b be any numbers and let m and n be positive integers. Then, our Properties of Exponents are as follows: 1. Product Property: (expanded form to simplified form) b m b n = b m+n e.g = Quotient Property: (expanded form to simplified form) b m b = n bm n e.g = A Power to a Power Property: (expanded form to simplified form) (b m ) n = b mn e.g.(6 2 ) 5 = Product to a Power Property: (simplified form to expanded form) (ab) n = a n b n e.g.(2 7) 3 = Quotient to a Power Property: (simplified form to expanded form) ( a ) ( ) n a n 4 7 = e.g. = 74 b b n **Let s use Frayer Models to better understand the properties above!** High School Math Prep 2 of 10 c Sa diyya Hendrickson

3 Understanding Properties I. Product Property (PP) II. Quotient Property (QP) High School Math Prep 3 of 10 c Sa diyya Hendrickson

4 Understanding Properties III. Power to a Power Property (P p P) IV. Product to a Power Property ((Pr) p P) High School Math Prep 4 of 10 c Sa diyya Hendrickson

5 Understanding Properties V. Quotient to a Power Property (Q p P) 1. Raising a number or variable to a positive exponent n means the repeated multiplication of the number/variable exactly n times. 2. If you re ever in doubt of a property, use the definition of repeated multiplication (i.e. write out the longer, expanded form of the expression as we did when we explained the properties with numbers) to verify that the one you are using is correct. 3. Remember that depending on the question, you may need to go from the expanded form of a property to the simplified form, or vice versa. Because the properties say equal to you can move in either direction! 4. These properties increase your list of rules to the game of math. The better you know these rules, the better you will be at playing this game! High School Math Prep 5 of 10 c Sa diyya Hendrickson

6 Zero and Negative Exponents Consider the following examples: 1. We know that: = 1, while on the other hand: = QP = 5 ( ) = 5 0. Therefore: 2. We also know that: 5 0 = = Therefore: reducing = 1 5 while on the other hand: = QP = 5 ( ) = = 1 5 (the reciprocal of 5!) The examples above show that the definition of repeated multiplication no longer applies to the zero and negative exponents. To get a better understanding of these exponents, let s complete the pattern below: High School Math Prep 6 of 10 c Sa diyya Hendrickson

7 Zero and Negative Exponents 1. zero exponent: b 0 = 1 for any number b as long as b 0. General explanation: 1 = bn b n QP = b (n n) = b 0 2. negative exponent: b n = 1 b n for any number b as long as b 0. General explanation: 1 b n = b0 b n QP = b (0 n) = b n by the definition of the zero exponent = b ( 1)(n) by definition of multiplying by 1 = b (n)( 1) by Commutative Property of Multiplication P p P = (b n ) 1 ( b n = 1 ) 1 So, the negative part of the exponent just tells us to flip the fraction. i.e. it creates the reciprocal of the original number! Positive exponents cause repeated multiplication! The zero exponent creates the number 1! Note: Exponents do not have the power to make a number equal 0. The negative part of an exponent creates the reciprocal of the given number (i.e. makes numbers flip)! Note: Exponents do not have the power to make a number negative. High School Math Prep 7 of 10 c Sa diyya Hendrickson

8 Fractions in Disguise Because negative exponents create reciprocals, a number raised to a negative power may be a fraction in disguise! Let s explore a few examples. ( ) (5 1 ) = 2 1 ( ) ( ) 2 1 = 1 5 = by definition of the negative exponent by the definition of fraction multiplication = (2 3 ) = ( 3 1 ) ( ) by definition of the negative exponent = by the definition of fraction multiplication = by equivalent fractions and LCD = 2 3 = by Product Property (i.e = 2 (2+1) = 2 3 ) = by definition of adding fractions = 5 8 already in reduced form! [ 10 (7 2 ) ] = 5 ( ) 7 2 = = = = 7 12 by definition of the negative exponent by definition of fraction multiplication & squaring by theorem of fraction division by reducible pairs: (5, 10) and (42, 49) by definition of fraction multiplication High School Math Prep 8 of 10 c Sa diyya Hendrickson

9 Let s Play! 1. Use Properties of Exponents to simplify the following: (a) (b) (c) (d) (7 5 ) 8 (e) (f) (g) m 2 n 3 m 5 n 7 (h) 15q5 r 3 q 2 3q 6 r 2 2. Use Properties of Exponents to expand the following: (a) (11 12) 2 (b) 5 (7 6) (c) 9 (9 7) (d) ( ) (e) 25 (3+6) 3. Use your rules to prove the following results. State each rule explicitly. (a) ( ) = 4 3 (b) 5 3 ( ) = Evaluate and express in simplified/reduced form: (a) 4(15 1 ) 16(5 2 ) (b) 3(4 1 ) + 4(3 1 ) (c) 4(3 2 ) 3 3 (d) ( ) 4 1 ( 12 ) (e) ( 3 4 ) 2 ( ) 1 Solutions: 1. (a) 3 9 (b) (2 5) 4 (c) ( 3 4) 6 (d) 7 40 (e) 13 3 (f) 7 4 (g) m 7 n 10 (h) 5qr 2. (a) (b) (5 7 ) 6 (c) (d) (e) Start with the left side and with every step taken to get to the right side, state the rules that you are using. 4. (a) 5 12 (b) (c) (d) 3 7 (e) 3 2 High School Math Prep 9 of 10 c Sa diyya Hendrickson

10 Let s Play! Powers Table Number Squared Cubed Fourth Fifth Sixth ,024 4, ,125 15, ,296 7,776 46, ,401 16, , ,096 32, , ,561 59, , ,000 10, ,000 1,000, ,331 14, ,051 1,771, ,728 20, ,832 2,985, ,197 28, ,293 4,826, ,744 38, ,824 7,529, ,375 50, ,375 11,390, ,096 65,536 1,048,576 16,777, ,913 83,521 1,419,857 24,137, , ,976 1,889,568 34,012, , ,321 2,476,099 47,045, , ,000 3,200,000 64,000, , ,481 4,084,101 85,766, , ,256 5,153, ,379, , ,841 6,436, ,035, , ,776 7,962, ,102, , ,625 9,765, ,140, , ,976 11,881, ,915, , ,441 14,348, ,420, , ,656 17,210, ,890, , ,281 20,511, ,823, , ,000 24,300, ,000,000 High School Math Prep 10 of 10 c Sa diyya Hendrickson