Chapter 4 Exponents and Radicals 4.1 Square Roots and Cube Roots 1. a) 81 b) 225 c) 625 d) 4 9

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1 Chapter Exponents and Radicals.1 Square Roots and Cube Roots 1. a) 1 5 c) 5 d) 9 e) 5 f ) ( 9 ). a) 79 7 c) 1 d) e) 1 f ) 15. a) 5 1 c) d) e) f ) 7 g) 1 h) x i) 7a 1b. a) c) 1 d) 0 e) f ) 5 g) 7 h) 5y i) 9a 5. a) perfect cube perfect cube V units s units c) both or A 15 5 units s 15 units s 5 units V 15 5 units d) perfect square A 19 units s 1 units 7 7 e) both V 51 units s units MHR Answers or

2 f) both A 5 units V 5 units 79 1 s 1 units 7 or A 79 units V 79 units 9 s 7 units s units s 9 units. a) perfect square perfect square c) both d) perfect square e) both f ) neither 7. a) perfect square perfect cube c) perfect cube d) perfect square e) perfect square f ) neither. a) 17 c) 1 d) e) 1 f ) 7 9. The storage container will measure 1. m by 1. m by 1. m (or 10 cm by 10 cm by 10 cm). 10. The side length of the patio is ft. 11. V s 1. s 1. s s The edge length of the cube would be approximately 1.9 cm. 1. ft ft 1. m m m mm 1. approximately. m m by 9 m by 9 m 1. 1 cm 19. a) y 0 y a) x x 1. Volume of the tank in cubic inches: 1 ft (1 in.)(1 in.)(1 in.) 17 in. 5 ft (5)(17) 9 1 The volume of the tank is equal to 9 1 in. Volume of one balloon: V _ πr V _ π() V π V The volume of one balloon is approximately 90.7 in. Number of balloons inflated per full tank: volume of tank volume of balloon _ V 10.1 A full tank will inflate approximately 10 balloons cm Mathematics 10 Exercise and Homework Book MHR 7

3 . a) _ _ _ _ c) Answers may vary. Look for the idea that a perfect decimal square exists if it has an even number of zeros before the perfect square number.. The expression _ 5 is not a perfect square because when you multiply two positive or two negative numbers the answer is always positive. The expression 7 is a perfect cube because when you multiply three negative numbers, such as ( )( )( ), the answer is negative. Therefore, it is possible to have a negative perfect cube. 5. a) When you double the side lengths of a square, the area increases by a factor of or. Example: A s (s) s When you triple the side lengths, the area increases by a factor of or 9. Example: A s (s) 9s When you double the edge lengths of a cube, the volume increases by a factor of or. Example: V s (s) s MHR Answers When you triple the edge lengths, the volume increases by a factor of or 7. Example: V s (s) 7s. Integral Exponents 1. a) 1 x _ 1 c) (5x) or 1 _ 5x d) a b e) 5 a f ) _ a b 5 g) ( ) h) x y _ i) a b. No. Shelby s answer is incorrect. The 1 x 10 correct answer is y.. a) c) 0.05 d) 1 e) 09 f ).77. a) a b b 5 a c) p 1 d) s 10 e) x f ) t 1 g) 1 n h) y x 5. a) () () ( ) ( ) ( ) ( ) ( ) 1 ( ) _ c) ( ) 5 d) ( 0 ) ( 0 ( ) ) ( ) e) ( ) ( )() 1

4 f ) ( ) () ()( ) () 1 _ 1 () 1 g) [( )( 7 )] [() +( 7) ] h) ( ) [() ] ( )( ) 9 _ ()()( ) ()()( ) i) (a ) () a ( )( ) () a a j) [( )( )] [() +( ) ] [() 1 ] (). a) 7 00 cm cm 7. approximately 1 caribou. a) 00 bacteria bacteria c) 50 bacteria 9. [(( 1 ) ) ] 1 [(( 1 ) [ ( 1 ) ] 1 [ ] ) ] Or, some students may evaluate as. 10. No. Kevin is incorrect. Example: Since the bases are not the same, you cannot add the exponents. When simplified, the expression ( )( ) ()(9) 7. The power a) 5 g 00 g 1. a) d 1 gt 1 (9.)(1. ) (.9)(15.7) 75. The penny falls from a height of approximately 75. m. d 1 gt.5 1 (9.)t.5 (.9)t _.5.9 t t _ t t It takes approximately. s for the penny to fall. c) d 1 gt 1 (9.)t (.9)t.9 t t 50.1 t 50.1 t 7.11 It takes approximately 7.1 s for the penny to fall. 1. a) approximately or 7. billion people approximately or. billion people 1. a) approximately or.5 million people approximately or 7.5 million people 15. a) A 0.01() 0.0 After years, the payment will be $0.0. A 0.01() After 10 years, the payment will be $10.. A 0.01() After 5 years, the payment will be $ Mathematics 10 Exercise and Homework Book MHR 9

5 Accept any reasonable justification. Examples: I would accept the double the money offer because it is worth more over time. I would accept the cash prize because it is immediate and I have few financial resources at the present time. c) Years 0 10 total $0.7; years 11 0 total $ ; years 1 5 total $ The total value over 5 years is $ a) 1.5 g approximately 1. g c) approximately 0. g 17. a) x x c) x d) x 1. a) approximately 1.05 g approximately 0. g 19. Yes. Example: When you multiply the exponents within each expression, both are equal to. 0. x + x + x + x 5 x ( ) 5 x 5 x x x or x + x + x + x 5 x () 5 x ( ) 5 x+ x + x 1. For + +, use the order of operations to evaluate each power and then add the resulting values: For ( )( )( ), since the powers have a common base, you can multiply by adding the exponents : Example: calculating student enrollment at schools in the community. a) You would use a positive exponent to predict enrollment in future years beyond the current year. You would use a negative exponent to calculate student enrollment in years before the current year.. Rational Exponents 1. a) a 15 y 5 c) x 0.9 or x 9 10 d) a 0. e) x or 1 x f ) 9 g) x 1 1 h) 10 a i) a 1.5 or a. a) 1 a 5 1 c) y d) 1 a 7 e) a 1.5 b or a b f) x 9 _ 15 g) y h) x 1 _ x 5 y 0. a) ( x ) q x x q x q q q ( x ) x ( x )(x q ) x 1 + q x 1 x ( x + q 1 q 1 + q 1 + q 1 )( x 1 ) x 1 50 MHR Answers

6 y c) y q y 11 1 y q y 11 1 q 11 1 q 11 1 q 11 1 q 1 q 1 q 1 y y 1 y d) (7x 1 ) (qx 1 ) ( x _ ) ( x 1 q 1_ ) x 1 x 1_ x _ + ( 1) q 1_ x 1_ x _ + ( ) q 1_ x 1_ x 1 _ q 1_ x 1 q 1_ x 1_ x 1_ q 1_ q (7x 1 ) (x 1 ) q _ x 1 e) (5 q ) ( q ) 15 7 ( 5q _ ) q 15 7 ( 5 ) q 15 7 ( 5 ) q _ ( 5 ) ( 5 ) q ( 5 ) q (5 ) ( ) a) 9 c) 1 d) 7 e) 5 x _ y f ) 5 5. a) c) d) e) 0.07 f).7. a) trout 109 trout c) 911 trout d) 1 trout 7. a) Error: A common denominator is needed to subtract exponents. a a 1 a 1 a a Errors: The negative exponent needs to be converted to a positive exponent. The expression is equal to, not. ( 1y ) 0.5 (1) 0.5 (y ) y _ 0.5 y( )( 0.5). a) $ $1. 9. a) 1.5 represents the growth rate; 1000 represents the starting population approximately bacteria c) approximately 1 bacteria 10. a) 5.00 million people.155 million people 11. a) A (0.5) t 0 (0.5) After 5 min, approximately 5.9 g remain. A (0.5) t 0 (0.5) 0.75 After h, approximately 0. g remain. c) A (0.5) t 0 (0.5) After 1 h, approximately 0.0 g remain Mathematics 10 Exercise and Homework Book MHR 51

7 1. a) Time (t) Concentration (C) Concentration Remaining (mg) C Medication Remaining in Bloodstream Time (h) 9 1 c) t h d) A mg g 1. a) $.1 $ x 1_ ( ) x 1_ () x 1_ + 1 x 1_ + x _ x x _. Irrational Numbers 1. a) ( 5 ) ( ) c) ( 5 ) d) 1 e) f) x h) ( x y ) [( 1 9) 1 5 ] ( 1 9 ) 5 g) ( a ). a) (5t) c) x d) ( a b ) 1 5 or a 5 b 5 e) y 5 f ) a t. a) 0.5 c) 10.9 d) 1.5 e).51 f ) a) 5 _ ( ) 5 _ (1)(5) _ 0 c) 5 d). e) 17. f ) 10 5 or 5 5. a) _ c) _ 750 d) _ 11 e) 5 f ) a) 11 c) 10 d) 5 e) 10 f ) a) 15 c) 1 d) e) 5 f ) 1. a) 0. _ 7,, 0.5 _, 0.9 ; 0.5 is an irrational number. 0.,, 0., 0. ; 0. is an irrational number. 9. a) and 5 are irrational numbers. _ 5 1 _ _ approximately cm 11. approximately 1.5 cm and 7 are irrational numbers. 1. V s (1.)(10 9 ) s (1.)(10 9 ) s 1091 s The edge length of a cube that contained Earth s estimated total volume of water would be approximately 1091 km. 5 MHR Answers

8 1. a) approximately.1 cm approximately 7. cm 1..7 s 15. approximately cm 1. a) solution B: 0.15 M solution A: 0.7 M 17. a) approximately 110 m approximately 01. m 1. SA π [h ( V πh ) + ( V π [( 5 5 π ) + ( π )] πh )] π[( ) + (.00 7)] π[ ] π[ ] The surface area of the cylinder is 97 m. 19. a) 5 c) d) _ _ a) 7 ( 7 ) ( 7 ) ) _ 5 ( [ ( 5 ) 1 1 ] 1 () 5 ( )( 1 ) 5 1 c) ( 1 ) 1 10 d) ( 5) 1 1. The expression x does not have a solution when x is negative. It is not possible to determine the even root of a negative number. Example: The expression x ( ) _ 7 has no solution.. The expression x always has a solution because when you raise a negative number to an even exponent, the result is always a positive number and then it is possible to take the cube root of the positive number.. a) Example: For all non-perfect squares, the calculator screen shows 9 decimals. Yes. Example: The square root of each non-perfect square is an irrational number since it cannot be expressed as a terminating or a repeating decimal. Chapter Review.1 Square Roots and Cube Roots 1. a) perfect square perfect cube c) perfect square d) perfect cube e) perfect square f ) both. a) 19 There is one group of and one group of 7. Therefore, the square root of 19 is ()(7) There are three equal groups of s. Therefore, the cube root of 51 is ()()() Mathematics 10 Exercise and Homework Book MHR 5