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1 REVIEW, pages For each pair of numbers, write two expressions to represent the distance between the numbers on a number line, then determine this distance. a) 1 8 and 1 4 b) 7.5 and and , or 17 8 The numbers are 1 7 units 8 apart on a number line. 7.5 (.75), or , and The numbers are units apart on a number line Arrange in order from least to greatest. a) 6, 24, -2 6, 96 Each radical has index 2. Write each mixed radical as an entire radical # 6 2 # # 6 4 # is negative so it has the least value. Compare the radicands of the other radicals: 24<54<96 So, from least to greatest: 2 6, 24, 6, 96 b) 5 8, 72 50, , 5 Each radical has index 2. Simplify each radical # a1 4 b Compare the fractions: 2 < 5 5 So, from least to greatest: 2 8 < ,,, Chapter 2: Absolute Value and Radicals Review Solutions DO NOT COPY. P
2 . Write each entire radical as a mixed radical, if possible. 4 2 a) b) # 8 # # Write the values of the variable for which each radical is defined, then simplify the radical, if possible. a) 16x b) 64x 2 16x ç when 16x» 0; that is, when x» 0. 16x 16 # x 4 x 64x 2 ç when 64x 2» 0. 64>0 and x 2» 0 So, 64x 2 is defined for x ç. 64x 2 64 # x 2 8x c) d) 4-64x 16x 6 Since the cube root of a number is defined for all real values of x, the radical is defined for x ç. 64x 64 # x 4x 4 16x 6 ç when 16x 6» 0. 16>0 and x 6» 0 So, 4 16x 6 is defined for x» x # x 4 # x 2 2x 4 x Identify the values of the variables for which each radical is defined where necessary, then simplify. a) b) 16x - 75x x 6 # 2 25 # 2 9 # The cube root of a number is defined for all real numbers. So, each radical is defined for x ç. 8 # 2 # x 125 # # x 2x 2 2x 5 x 2x 5 2x 5 x P DO NOT COPY. Chapter 2: Absolute Value and Radicals Review Solutions 41
3 6. A square with area 75 square units has a square corner of area 27 square units moved as shown. Determine the perimeter of the resulting shape. Describe the steps you took to solve the problem. 5 5 The side length of a square is the square root of its area. So, the side length of the square with area 75 square units is: 75 5 units The side length of the square with area 27 square units is: 27 units Label the diagram. Perimeter of shape formed 5( ) 5 (5 ) 15 5 (2 ) 26 The perimeter of the shape formed is 26 units Identify the values of the variable for which each expression is defined where necessary, then expand and simplify. a) ( 5( 5 7) 5 7) b) 12 a + b21 a + b2 The radicands cannot be negative, so (2 a b( 2 a( b)( a» 0 and b» 0. a b) a b) a b) 2a 2 ab ab b 2a ab b 42 Chapter 2: Absolute Value and Radicals Review Solutions DO NOT COPY. P
4 8. Rationalize the denominator. 5-7 a) b) 5 ( 5 7) # 5 5 # 7 # 5 # # # 2 4 # # Simplify a) b) ( 7 5) 2 6( 7) 2 6( 5) ( 7) 2 ( 5) # ( 7 5) ( 7 5) ( 5 4 ) (6 2 ) # (6 2 ) (6 2 ) 5(6 2 ) 4 (6 2 ) (6 2) 2 ( ) ( ) Identify the values of the variable for which each expression is defined, then expand and simplify. a) 2 a1 b + a2 2 The radicands cannot be negative, so a» 0 and b» 0. 2 a( b a) 2 2 a( b a)( b 2 a[( b)( b a) a) a( b a)] 2 a[9b ab ab a] 2 a[9b 6 ab a] 18b a 12 a 2 b 2a a 18b a 12a b 2a a P DO NOT COPY. Chapter 2: Absolute Value and Radicals Review Solutions 4
5 b) 1 x + 2 y2 2-1 x - 2 y2 2 The radicands cannot be negative, so ( x 2 y) x» 0 and y» 0. 2 ( x 2 y) 2 ( x 2 y)( x 2 x( x 2 y) y) ( x 2 y)( x 2 y) 2 y( [ x( x 2 y) x 2 y) 2 y( x 2 y)] 9x 6 xy 6 xy 4y [9x 6 xy 6 xy 4y] 9x 12 xy 4y 9x 12 xy 4y 24 xy Determine the root of each equation. Verify the solution. a) 5 = 2x + 7 b) 1-2 x = 4 - x 2x 7» 0; that is, x» x ( 2x 7) x 7 2x 18 x 9 x» 0; that is, x» x 4 x x ( x) 2 2 x 9 x 12. Which equations have real roots? Justify your answers. a) 2 x + 5 = 5x - 11 b) 11x = x 5» 0; that is, x» 5 5x 11» 0; that is, x» 11 5 So, for both radicals to be defined, x» x 5 5x 11 (2 x 5) 2 ( 5x 11) 2 4(x 5) 9(5x 11) 4x 20 45x x x 119, or Since x 2 7 lies in the set of 41 possible values for x, the equation has a real root. 11x 2» 0; that is, x» x x 2 5 The left side of the equation is greater than or equal to 0. The right side of the equation is negative, 5. So, no real solutions are possible. The equation has no real roots. 44 Chapter 2: Absolute Value and Radicals Review Solutions DO NOT COPY. P
6 1. The approximate speed at which a tsunami can travel is given by the formula S = 9.8d, where S is the speed of the tsunami in metres per second, and d is the mean depth of the water in metres. A tsunami is travelling at 6 m/s. What is the mean depth of the water to the nearest metre? S 9.8d 6 9.8d (6) 2 ( 9.8d) 2 Substitute: S d d d The mean depth of the water is about 12 m. P DO NOT COPY. Chapter 2: Absolute Value and Radicals Review Solutions 45
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