Using exponents to decribe numbers

Math 9 UNIT 3 POWERS & EXPONENTS (Pg. 88-123) Key Words: Power Base Exponential Form Coefficient Exponent Homework Assignments: 3. 1 USING EXPONENTS to DESCRIBE NUMBERS (pg. 92 - 98) Homework: Pg. 97 - 98 3. 2 EXPONENT LAWS (pg. 99-107) Homework: Pg. 106-107 3. 3 #1-26 ORDER of OPERATIONS (pg. 108-113) Homework: Pg. 111-113 3. 4 #1-23 # 5-19 USING EXPONENTS to SOLVE PROBLEMS (pg. 114-119) Homework: Pg. 118-119 #3-13 Review: Chapter 3 Review: Pg. 120-121 Chapter 3 Practice Test: Pg. 122-123 1 Math 9 x Unit 3 - Challenge Questions: 1. The brightness of a light source appears to decrease as distance, d, increases. If brightness decreases by 1 d2 , how does the brightness change if the distance is doubled? 3. a) Evaluate 762. The square of 76 ends in 76. b) What other two-digit number squared also ends in itself? 1 2. Consider the expression x x . If x is a fraction, predict what increasing the value of x does to the value of the expression. Explain your thinking. Test your prediction with a calculator. 4. A box measures 2 m by 3 m by 4 m. What is the length of the longest straight pole you could place in the box? 2 Math 9 3.1 USING EXPONENTS to DESCRIBE NUMBERS (pg. 92 - 98) A power is: repeated multiplication M Pg. 93 t A base is: base -> getsrepeatedly y multiplied An exponent is: oftimes youmu't the base exponent y Exponential form is: 3^5 or 35 35 or X The 5 is theexponent _____________ The 3 is the base _____________ For 45 the power is _________ 45 4 the base is _________ 5 the exponent is _________ ______ 54 5 to the power of 4 625 The exponent is the number of times the base is multiplied by itself: 23 = 2 x 2 x 2 = 8 53 = 5 5 5 125 14 = 1 1 1 1 1 31 = 33 9 35 = 3 3 3373 243 = 3 Math 9 We can also work backwards: 25 = 52 32 = 25 16 = 42 49 = = 25 =2 or 72 2 272 2 24 7 7 Standard form is the way most people would talk about numbers. Most people would to the power of 3 (103), they would say 10 x 10 x 10 = 1000. Ex. 100, 32, 25, 16 Exponential form is the base number written to an exponent. 52 24 Ex. 102, ___, 25 ___, ___ Ex. Write the following numbers in exponential form: 53 128 ______ 27 125 _______ 35 34 _______ 92 or _______ 243 _______ or _______ no 81 Ex. Write the following numbers in standard form: 24 ________ 2222 35 3 _________ 3 33 3 63 _________ 66 6 We must be careful about negatives. If the negative is within a bracket, the exponent is for the negative sign as well. -42 _______ = ________ 16 44 (-4)2 _______ 16 positive 4 4 = ________ -33 _______ 27 3 3 3 = ________ (-3)3632 ______ 27 3263= _______ t appearsON 4 Math 9 Name:__________________ Ariane 3.1 Worksheet 1. Write each expression as a power. Then, evaluate. Power Evaluate 43 ________ a) 4 × 4 × 4 ________ b) 2 × 2 × 2 × 2 × 2 × 2 ________ 641 zor 64N ________ 2. Write each expression as a power. Identify the base and the exponent in each power. Then, evaluate. Power Base Exponent Evaluate a) 1 × 1 × 1 × 1 × 1 × 1 × 1 ____ 17 ____ b) 5 x 5 x 5 x 5 x 5 ____ ____ 55J 1 5 7 I ____ ____ 311251 ____ ____ 5 3. Write each power as repeated multiplication. Then, evaluate. Repeated Multiplication a) 25 6 b) 10 c) d) e) Evaluate 32 212 2 2 2____________________ ______ ____________________ 10710 10 101010 V ______ 1000000 V 12 2 2 2 24 ____________________ 43 444 V 43 ____________________ ( 6)4 ____________________ C6 C6 CG C G V 24 5 6zeroes 106 ______ 16 64 ______ 12961 ______ 5. Complete the table. Repeated Multiplication Exponential Form a) ( 3) × ( 3) × ( 3) × ( 3) 13 275 b) ( 2) × ( 2) ×( 2) ×( 2) ×( 2) c) d) Aodd ever 6 C6 x C5 5 621 6 6 6 ( 6)5 C5 3 C5 ofregsmultipliedtogethermakesanswerneg positive Value 81 32 7776 125 5 Math 9 1. Fill in the blanks to rewrite each formula using repeated multiplication. Then, answer the questions. a) S.A. = 6s2 6 S S.A. = ____ × ____ S × ____ What does S.A. stand for? ____________________ surface area What solid is this formula used for? ____________________ cube SA of b) V = r2h h V = ____ it × ____ r × ____ r × ____ What does V stand for? ____________________ volume What solid is this formula used for? ____________________ volume of cylinder 2. Fill in the blanks to rewrite each formula using repeated multiplication. Then, answer the question. a) s2 = ____ S S × ____ area What shape or solid is this formula used for? ____________________ b) square 2 r = ____ Tu × ____ r × ____ r areashape or solid is this formula used for? ____________________ What circle 2 h c) 2s2 × h = ____ × ____ s × ____ S × ____ volume What shape or solid is this formula used for? ____________________ prism 6 Math 9 3.2 EXPONENT LAWS (pg. 99-107) Operations with Exponents: If we want to multiply numbers that have exponents and the same base: 32 x 34 = = 3 3743 3 33 36 When we multiply numbers with the same base we can add the exponents: 23 x 25 = = 322 28 (-5)4 (-5)3 = S = C5715 C5 C5 C5 C5 s In general terms we can write (am)(an) = _____________ amin If we want to divide numbers that have the exponents and the same base: 45 = 42 YII = 43 When we divide numbers with the same base we can subtract the exponents: 68 = 1119 655 65 bottom 0 = 63 ofb'sat out ontop cancels ( 3) 2 = ( 3)1 433537 = 3 or C3 something am In general terms we can write = am - n : an 7 Math 9 If we have an exponent to the exponent: sets I (3 ) = 2 4 3 36 3 = = 8three's When we have an exponent to the exponent we can multiply the exponents: ADVANCED 444 (43)5 = = aah aah aan [(-4) x 2]3 = 444 4 271 4 27 42 = 4 s 431 43,412302302 In general we can write (am)n = _____________ C4731233 theorderyoumultiplydoesnotmatter ifyouhaveaformulawithtiproation m m m We can also write (a x b) = a x b extension opportunity Ex. Write [(-4)3]3 as a single power. Evaluate. C4 C4 C DC DC DC477 454 4 4 4 433 or C4 9 Ex. Write (-3 x 2)2 as the product of two powers. Evaluate. 1 33 3 C332 Ex. Write 2 5 m 735124 4 as the quotient of two powers. Evaluate. am ya Note: Anything to the power of 0 is 1! 24300= I (-567)0 = -42500= g I d I E I mm 0 top bottom subtractexponents 1 8 Math 9 3.2 Worksheet 1. Name Ariane __________________ Write each expression as a single power. Then, evaluate. Single Power 28 a) 24 × 24 ______ b) ( 4)2 × ( 4)2 ______ 1D c) 62 × 6 ______ 63 ______ 96 d) 93 × 93 Evaluate ______ My ______ ______ ______ 2. Write each expression as a product of two powers, then as a single power. Product of Two Powers Single Power 347133 a) (3 × 3 × 3 × 3)(3 × 3) ____________ b) (5 × 5 × 5 × 5)(5 × 5 × 5 × 5 × 5 × 5) ____________ 54 59 ____________ 86 85 c) (8 × 8 × 8 × 8 × 8 × 8)(8 × 8 × 8 × 8 × 8) 2 113 ____________ 1 d) (11 × 11 × 11)(11 × 11) 36 ______ 5 ______ 8 ______ 115 ______ 3. Write each expression as a single power. Then, evaluate. Single Power 3 ______ ______ to ______ ______ 2 Z ______ I ______ a) 34 ÷ 32 ______ b) ( 5)3 × ( 5)2 c) ( 2)2 d) 82 ÷ 82 3 80 Evaluate 6 ______ 9 Math 9 4. Write each expression as a quotient of two powers, then as a single power. Quotient of Two Powers Single Power 54 5 73 73 a) (5 × 5 × 5 × 5) ÷ (5 × 5) ____________ b) (7 × 7 × 7) ÷ (7 × 7 × 7) ____________ 8 8 8 8 8 8 8 8 8 8 8 ____________ Ii 87 (2 2 2 2 2 2) d) (2 iii 2 2 2I 2) z c) 87 84 26 25 ____________ g ______ 70 I ______ 83 ______ 2 ______or 2 5. Complete the table. Expression a) 3 4 2 Repeated Multiplication 3 14 4×4×6×6 b) (4 × 6)2 c) 2 3 14 5 2 2 2 2 2 383 3 3 3 3 Two Powers 32 × ( 4)2 42 62 25 35 6. Does 82 =( 8)2? Explain your answer. No because an odd ofregsmultiplied togethermakes an answer negative Aneven ofregsmultiplied together makes an answer positive Therefore 64 C85 64 82 10 t FOI least Yffolly Laws of Exponents: a a In x X 3 7 X E E E x X X x 7 X X a 1 E 3 Math 9 3.1/3.2 Quiz Name:__________________ 1. Write each expression as a power. Identify the base and the exponent in each power. Then, evaluate. Power Base Exponent Evaluate a) 5 × 5 × 5 ____ ____ ____ ____ b) 7 × 7 × 7 × 7 × 7× 7 ____ ____ ____ ____ c) 6 x 6 ____ ____ ____ ____ 2. Write each power as repeated multiplication. Then, evaluate. Repeated Multiplication Evaluate a) 63 ____________________ ______ b) ( 4)3 ____________________ ______ ____________________ ______ c) ( 6)3 3. Complete the table. Repeated Multiplication Exponential Form Value a) ( 2) × ( 2) × ( 2) × ( 2) b) ( 3)5 c) 625 4. Write each expression as a single power. Then, evaluate. Single Power Evaluate a) 34 × 34 ______ _________ b) (-1)2 × ( 1)2 ______ _________ c) 52 × 5 ______ _________ d) 143 × 142 ______ _________ 11 Math 9 5. Write each expression as a product of two powers, then as a single power. Product of Two Powers Single Power a) (-3 × -3 × -3 × -3)(-3 × -3 x -3) ____________ ______ b) (2 × 2 × 2 × 2)(2 × 2 × 2 × 2 × 2) ____________ ______ c) (18 × 18 × 18 × 18 × 18)(18 × 18 × 18 × 18 × 18) ____________ ______ 6. Write each expression as a single power. Then, evaluate. Single Power Evaluate a) 23 ÷ 22 ______ ______ b) ( 5)3 × ( 5)2 ______ ______ ______ ______ ______ ______ c) ( 2)2 3 d) 122 ÷ 122 7. Write each expression as a quotient of two powers, then as a single power. Quotient of Two Powers Single Power a) (-2 × -2 × -2 × -2) ÷ (-2 × -2) ____________ ______ b) (8 × 8 × 8 x 8) ÷ (8 × 8 × 8) ____________ ______ ____________ ______ c) 8 8 8 8 8 8 8 8 8 8 8 8. Does 22 x 33 = 65? Explain your answer. 12 Math 9 3.3 ORDER of OPERATIONS (pg. 108-113) Order of operations always applies! We know the terms A coefficient is a power with a number multiplied in front of it. 38 For -2(38), __________ is the power __________ is the exponent __________ is the base 3 __________ is the coefficient 2 Simplify then evaluate Ex. 5 x 43 = 5 43 Ex. (-3)(-4)2 C 3 16 5 64 320 48 cut 2 Ex. -9 + (-4)3 y g 64 145 92 13 t A coefficients ordercf operations Math 9 Ex. 33 (5)(-3)2 + 6 6 5 9 3 to extendskills 27 5 a 6 27 45 6 Ex. -4(-12 + 52) + 3(-1 + 5)2 55 25 3 C 1 4 4 137 314 12 12 D D 25 3 4 2 3 16 48 256 Ex. -2(53 + 26) 24 ÷ 23 2 125 64 24 8 2 1897 3 378 3 3750 Ex. 12 4 13 52 Ex. 43 + -44 64 4 3 4 2 C 125 64 24 3 2 189 3 378 3 2 x [5 - (-2)2]3 3812 0.752 X 5433 0.5625 1 0.56250 Ex. 2 5 0.44 4 ÷ 25 8 2 3.1252 0.0256 9.765625 0.002621440 14 Math 9 Ariane 3.3 Worksheet 1. Name __________________ Evaluate each expression. Show your work. a) 4(HgÉEnat 24) 4116 isharming b) 7(43) IT s64 64 2. Write each expression, using a coefficient and a power. Then, find the value of each expression. Show your work. a) 5 × ( 2) × ( 2) × ( 2) × ( 2) × ( 2) II b) 6( 10)( 10)( 10)( 10)( 10) 6 1105 8888804 3. Evaluate using your calculator. a) 4( 9)3 b) 74 2401 2916 4. Evaluate. Show your work. a) 12 + ( 4)2 2 ( 33) 16 427 Yg 1 27 c) 36 52 + (43 62) 36 25 64 11 28 39 361 b) 53 4( 26) 1 125 4664 13285 256 d) ( 32) + ( 9)2 1 97 81 90 15 Math 9 3.4 USING EXPONENTS to SOLVE PROBLEMS (pg. 114-119) There are many problems that involve exponents, for example population growth. Ex. One type of bacteria double in population every 1 hour. If you touch a door knob with one bacterium and then touch your mouth, draw a table of the number of bacteria in your system over the course of 24 hours. Time (hours) # of times population has doubled Bacteria Population 0 hr 0 1 Population as a power of 2 Time (hours) # of times population has doubled Bacteria Population Population as a power of 2 1 hr 2 hr Try to come up with an expression in exponential form to determine the number of bacteria after x hours. Explain what each value represents 16 Math 9 Now lets look at what happens over 12 hours if our initial bacteria population in your body is 3 bacterial cells that you touched on the door handle. Time (hours) # of times population has doubled Bacteria Population 0 hr 0 3 Population as a product of 3 and a power of 2 1 hr 2 hr Try to come up with an expression in exponential form to determine the number of bacteria after x hours. Explain what each value represents. How many hours would it take for the population to grow to 786,432? 17 Math 9 When we work with formulas with exponents we should follow the same steps: 1. 2. 3. 4. Write down the formula to be used Substitute the known values Solve for the unknown value Answer with units! Ex. A right triangle has a hypoteneuse with side length of 45m2. If one leg is 3m, how long is the other? Ex. A body is thrown off the Port Mann bridge. The height of the body in metres, h, at time t, in seconds, is given by h = -4.9t2 + 28t What is the height of the body after 2 seconds? What is the height of the body at its highest point (this occurs at 2.9 seconds)? 18 Math 9 3.4 Worksheet Name_______________ 1. What is the volume of a cube with a side length of 4 cm? Show your work. 2. A colony of bacteria triples every hour. There are 30 bacteria now. How many will there be after each amount of time? Show your work. a) 1 h b) 3 h c) 12 h d) n h 3. What is the surface area of a cube with a side length of 6 cm? Show your work. 4. The diagram shows a circle inscribed in a square with a side length of 16 cm. What is the area of the shaded region? Give your answer to the nearest hundredth of a square centimetre. Show your work. 19 Math 9 3.3/3.4 Quiz 1. Name __________________ Evaluate each expression. Show your work. b) 6( 5)2 a) 5(3)3 2. Write each expression, using a coefficient and a power. Then, find the value of each expression. Show your work. a) 4 × 3 × 3 × 3 c) 1 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 3. Evaluate using your calculator. a) 6(4)6 b) 7 × 83 4. Evaluate. Show your work. a) (6 + 3)2 c) [(9 21 b) 62 52 ( 2)]2 + ( 3)3 d) 12 3(42) 20 Math 9 5. Bacteria reproduce by splitting in two. If a single bacteria divides every 30 min, how many bacteria will a single bacteria produce after 12 h? a) Write the answer in exponential form. b) Calculate the answer. 6. In the formula, d = 4.9t2, d is the total distance, in metres, and t is the time, in seconds, that the skydiver free falls. Calculate the distance the skydiver falls in the following times. Show your work. a) 2 s b) 4 s 7. Find the side length of the square attached to the hypotenuse in the diagram. Show your work. 21 Math 9 Chapter 3 Test Name ______________ Per___ Multiple Choice 1. In the equation ( 2)5 = 32, which number represents the base of the power? A) 32 B) 2 C) 1 D) 2 2. Which expression is equivalent to ( 2) × ( 2) × ( 2) × ( 2) × ( 2)? A) 25 B) 32 C) ( 2)5 D) ( 2)5 3. What is the product of 52 and 54? A) 650 B) 256 C) 58 D) 56 4. Devin was asked to simplify the expression 10 10 = 10 = 10 = 10 = 14 23 × (3 20)2 6 × (3 1)2 6×4 24 23 × (3 20)2.His work is shown below. Step 1 Step 2 Step 3 Step 4 In which step did Devin make his first mistake? A) Step 1 B) Step 2 C) Step 3 D) Step 4 5. Evaluate the expression -(-3)3 A) 9 B) -9 C) 27 D) -27 22 Math 9 6. Evaluate 33 A) 75 3(-2)4 B) -21 C) 51 D) 3 7. Two students were asked to write each product of powers as a single power. Their work is shown below. Danica Frank 33 × 32 = (3 × 3 × 3) (3 × 3) = 35 33 × 3 2 = 3 3 × 2 = 36 Which of the following statements about their procedures is true? A) B) C) Both Danica and Frank have no errors in their procedure. D) Both Danica and Frank have errors in their procedure. Short Answer (SHOW ALL YOUR WORK) 8. Evaluate 33 + 30 9. The expression 5 10 3 written as a fraction in simplified form is _____. 10. Arrange the powers in order from smallest value to largest value. ( 4)2, (2)3, (4)3, ( 1)5 23 Math 9 11. Write each expression as a single power in simplified form. a) 67 ÷ 64 = __________________________ b) (22 + 3)4 = __________________________ 3 c) 24 = __________________________ 12. Complete the table below: (12 marks) Expression Repeated Multiplication As a power(s) Evaluate (-7)2 6x6x6x6x6 (-5)3 243 -32 [3 x (-2)]3 (8 x 7) x (8 x 7) 3 4 3 3 3 3 3 ( 2) ( 2) ( 2) ( 2) 24 Math 9 13. Does (-6)4 = -64? Explain your answer! (2 marks) 14. Finish the following statement: (1 mark) am x an = ______________ 15. What is a coefficient? Give an example. (2 marks) 16. If 3x = 11, use exponent laws to evaluate the following expressions (show your work!) a) 32x b) 3(x + 1) 17. Find the value of the following expressions (SHOW ALL WORK!) (3 marks each) a) 24 2 2 + (7 2 2 5 ) (2)6 (2) 2 13 20 b) ( 1 22 )5 25 Math 9 Extended Response 18. For every metre you dive below the water surface of a lake, the light intensity is reduced by 5%. The percent of light intensity can be represented by I = 100(1 0.05)d, where I is the intensity of light, as a percent, and d is the depth, in metres. The intensity of light at the surface of the lake is 100%. Austin wanted to determine the light intensity at a depth of 3 m. His solution is shown below. I = 100(1 0.05)d I = 100(1 0.05)3 I = 100(13 0.053) I = 100(0.999875) I Austin realized that it is not possible for the light intensity to be approximately 100% at a depth of 3 m. a) light intensity at a depth of 3 m. Give your answer to the nearest whole percent. b) What is the light intensity at a depth of 15 m? Give your answer to the nearest whole percent? 26 Math 9 19. The formula to calculate the value of an investment growing annually with compounded interest is A = P(1 + R)n Where A is the accumulated value, P is the principal, R is the annual interest rate in decimal form, and n is the number of years that the investment has been growing. a) An amount of $2000 is invested at an annual interest rate of 6%. What will its value be after 3 years? b) How many years will in take for the investment to double in value? 27