Chapter 4: Radicals and Complex Numbers
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1 Chapter : Radicals and Complex Numbers Section.1: A Review of the Properties of Exponents #1-: Simplify the expression. 1) x x ) z z ) a a ) b b ) 6) 7) x x x 8) y y y 9) x x y 10) y 8 b 11) b 7 y 1) y 1) 1) 1) 16) 17) (x ) 18) (y ) 19) (z ) 0) (b ) 1) (xy ) ) (xyz ) ) (x y) ) (xy ) ) ( x y) 6) ( xy ) 7) xy x y 8) ab a b 9) xx x 0) y y y 8 1) (ab ) (ab ) ) ( xy ) ( xy) ) (p q ) (pq) ) (x y) (xy ) ) 1x y 6) 1x y 7x y xy 7) ( x ) 8) ( y ) 9) ( x y ) 0) ( x y ) 1) ( xy z ) ) ( x yz ) 8
2 Chapter : Radicals and Complex Numbers Section.: A Review of the Properties of exponents (the power of 0 and negative exponents) #1-1: Simplify the expression. 1) x 0 ) y 0 ) 0 ) 0 ) - 0 6) - 0 7) (-) 0 8) (-) 0 9) c 0 10) b 0 11) (x) 0 1) (ab) 0 #1-8: Simplify the expression. Write the answer with positive exponents only. 1) 1) 1) x 16) b 17) 1 18) 1 19) 0) a w x z 6 1) y ) x ) xy ) x y ) ( ) 6) ( ) 7) ( x y ) 8) ( x ) 9) x x 6 0) y y 1) x x ) y y ) x x z ) z 6 x ) x z 6) z 7) x y 6 1x y 8) xy 6x y 9
3 Chapter : Radicals and Complex Numbers Section.: Definition of nth Root #1-0: Evaluate the roots.. 1) 9 ) 16 ) 81 ) 6 ) 9 6) 7 7) 8 8) 7 9) 8 10) 1 11) 8 1) 7 1) 1 1) 1 1) ) 9 17) ) ) 16 0) 81 1) 16 ) 81 ) 16 ) 81 ) 9) 7 6) 0) 6 7) 8) 9 #1 - : Use a calculator to evaluate the expression, round to four decimal places. 1) 1 ) 6 ) ) ) 77 6) 1 7) 10 8) 0 9) 6 0) 0 1) 7 ) + 6 ) Simplify a) b) ( ) c) x ) Simplify a) b) ( ) c) y ) Simplify a) 6) Simplify a) b) ( ) b) ( ) c) x c) x 0
4 Chapter : Radicals and Complex Numbers Section.: Definition of nth Root #7-: Simplify the radical expressions. Use absolute values when necessary. 7) a 8) b 9) y 0) z 1) a 6 ) b 6 ) y 7 ) b 7 # - 78: Simplify the expressions. Assume all variables are positive real numbers, so no absolute values will be needed in any of the answers. ) x 6) 16b 7) x 6 8) y 10 9) z 8 60) x 1 61) x 6 6) 16z 10 6) 6x 0 6) 6x 1 6) y 1 66) 7y 9 67) y 0 68) 16z 69) 6 y 8 70) 9x ) 8x9 y 1 z 1 7) 6x y 8 z 1 7) 9 16x 9x 7) 9y 7) 1x9 6z 6 76) 16 81y z 1 77) 1 9x 78) 1 x 1
5 Chapter : Radicals and Complex Numbers Section.: Definition of nth Root 79) Use a calculator to complete the table, round to two decimal places when needed. Sketch a graph of the function and find the domain and range of the function in interval notation. Let h(x) = x x h(x) 80) Use a calculator to complete the table, round to two decimal places when needed. Sketch a graph of the function and find the domain and range of the function in interval notation. Let h(x) = x x h(x) 81) Use a calculator to complete the table, round to two decimal places when needed. Sketch a graph of the function and find the domain and range of the function in interval notation. Let h(x) = x x h(x)
6 Chapter : Radicals and Complex Numbers Section.: Definition of nth Root 8) Use a calculator to complete the table, round to two decimal places when needed. Sketch a graph of the function and find the domain and range of the function in interval notation. Let h(x) = x + x h(x) #8-9: Create a table of values and sketch a graph of the function. You may use your calculator and should round to two decimal places when needed. Use the graph to find the domain and range of the function in interval notation. 8) a) f(x) = x b) f(x) = x 8) a) f(x) = x b) f(x) = x 8) a) f(x) = x + b) f(x) = x + 86) a) f(x) = x + b) f(x) = x + 87) a) f(x) = x + b) f(x) = x 88) a) f(x) = x 1 b) f(x) = x 89) a) f(x) = x + + b) f(x) = x + 90) a) f(x) = x + b) f(x) = x + 91) a) f(x) = x b) f(x) = x 6 9) a) f(x) = x 1 + b) f(x) = x
7 Chapter : Radicals and Complex Numbers Section.: Rational Exponents For this exercise set, assume that all variables represent positive real numbers unless stated. #1-6: Write the expression in radical notation, (do not simplify). 1) ) ) (x) 1 ) (y ) 1 ) ( y ) 6) ( xy 7 z ) #7-1: Write the expression using rational exponents rather than radical notation, (do not simplify) 7) x 8) y 9) x 10) y 11) a 1) x #1-0: Write the expression using positive exponents and radical notation, then simplify. 1) ) ) ) 1 17) 7 18) 6 19) 9 1 0) 8 1 1) ) 9 ) 6 1 ) 8 ) ) 1 1 7) ( 9 ) 8) ( 8 7 ) 9) ( 9 ) 0) ( ) #1 - : Simplify the expression using the properties of rational exponents. Write the final answer using positive exponents. 1) ) x ) 1 ) y ) (x 1 ) 6) (y ) 7) ) x y 9) ) x 1 x a a 1 0) y 6y ) y 1 y 1) x1 x ) b b # - : Use a calculator to approximate the expressions and round to decimal places. ) 1 6) 1 7) 8) 9 9) 0) 1) 6 1 )
8 Chapter : Radicals and Complex Numbers Section.: Properties of Radicals For this exercise set, assume that all variables represent positive real numbers unless stated. #1-8: Use the multiplication property of radicals to multiply the expressions. Then simplify the result. 1) x 8x ) x x ) ab a 8 b ) xy x y 9 ) x x 6) 6x 10x 7) 18x 8) x 1x #9-16: Use the division property of radicals to divide the expression. Then simplify the result. 9) 1x x 10) y 7 11) x y x 1) x x 9 1) 8x x 1) 16x 1) x x 16) x 7 x 9 x 1 #17 - : Simplify the radicals. 17) 0 18) 19) 0) 106 1) 6 ) 98 ) ) 96 ) 00 6) 00 7) 96 8) 86 9) x 0) y 7 1) a 7 ) b 11 ) x 10 ) w 7 ) x 6) x 7) y 1) y 0 8) y ) b 1 9) x 6 0) y 6 ) 0ab ) 7a b c ) 8x 6 y z 6) xy 6 z 7 7) xy 8) 10x 7 9) x 16y 0) 106 b 8 1) 81xy 8 ) 108 z 1
9 Chapter : Radicals and Complex Numbers Section.6: Addition and Subtraction of Radicals For this exercise set, assume that all variables represent positive real numbers unless otherwise stated. #1-8: Add or subtract the radical expressions if possible. 1) + 7 ) x + x ) ) 7 ) ) 7) x + 6 x 8) y + y + y 9) x x + x x 10) x y 7x y 11) 1 x x 1 x x 1) y y y y 6 1) ) ) 16) 8 17) x + x x 18) y y y 19) x y + x xy 0) a b ab ab a 1) 7x + 8x ) 16x + 81x ) ) 0x + 8x ) x x + x 6) y y y 7) x x + x x 8) x x 16x + x 6
10 Chapter : Radicals and Complex Numbers Section.7: Multiplication of Radicals For this exercise set, assume that all variables represent positive real numbers unless otherwise stated. #1-6: Multiply the radical expressions. 1) 1 ) ) 6 1 ) 10 1 ) 1b 0b 6) 8x 10x 8 7) ( 10)( 1) 8) (7 )( 6) 9) ( 1a )(a 10a) 10) ( 6x)( 1x ) 11) 10 ( + ) 1) 1 ( 7 ) 1) ( 6) 1) ( 7 1) 1) ( + )( ) 16) ( 6)( + 6) 17) ( 7 + )( 7 ) 18) (9 + 6)( 6) 19) ( x + )( x ) 0) ( + x)( x) 1) ( + )( 6) ) ( x + 7)( x 8) ) (6 + 6)( + 8) ) ( + 10)( + + 7) ) ( x)( + x x) 6) ( + x)(x x + 1) #7-6: Multiply the special products. 7) ( )( + ) 8) ( + 6)( 6) 9) (x + 7)(x 7) 0) (y x)(y + x) 1) ( + )( ) ) ( 6 + )( 6 ) ) ( + 6) ) ( 7) ) ( x) 6) ( + x) 7
11 Chapter : Radicals and Complex Numbers Section.8: Rationalization For this exercise set, assume that all variables represent positive real numbers unless otherwise stated. #1 - : Rationalize the denominator. 1) ) 7 ) 6 ) 10 ) x 6) 10x 7) 6 b 8) x x 9) a a 10) b b 11) 1) 8 x 6x 1) 1) 1) a 7 b 16) 17) 18) 6 19) 9 0) x x 1) 1 ) # - : Rationalize the denominators by multiplying by the conjugate. ) + ) 6+ ) 6) ) 8) 8+ 9) x+ y 0) 8 x 1) + ) 6+ + ) + )
12 Chapter : Radicals and Complex Numbers Section.9: Radical Equations #1-6: Solve the equation. Be sure to check your answers. If a solution is extraneous, say so in your solution. 1) x = ) y = 6 ) x + = 9 ) y = ) x = 6 6) y = 1 7) b 1 = 8) x + = 9) x 10) y 1) x 1 16) x + 7 = 11) x = 1) x + = 6 1) x = = = 1) x = 17) x 1 = 6 18) y 1 = 19) z 1 = 1 0) a 1 = 1 1) (x ) 1 = ) (x ) 1 = ) (x 1) 1 = ) (x ) 1 = ) + x = 6 6) x = 7) + x = 1 8) + 6 x = 9 9) x = x + 8 0) x = x + 1) x = x + ) x + 0 = x ) x + = x = ) x + 1 = x + 7 ) n = n + 6 6) x + 7 = x 8 7) x + = x 8) x + 1 = x 9) x 1 = x 0) x + = x 1) t + 1 = t ) t 7 t = 1 ) b 7 + b = 7 ) b b = ) y = y + 1 6) x + 7 = x 1 9
13 Chapter : Radicals and Complex Numbers Section.10: Complex Numbers #1-16: Simplify the expressions. 1) 16 ) ) 81 ) 9 ) 0 6) 7) ) 9 9) 18 10) 1 11) 10 1) 6 1) 0 1) 1) 16) 8 6 #17 - : Perform the indicated operation, write your answer in standard form. 17) (6 + i) + ( i) 18) ( i) + ( 7i) 19) ( 6i) - ( i) 0) (8 + 9i) (7 9i) 1) ( + 6i) + ( i) ( + i) ) (-i) (-i) ( i) ) (i)(i) ) (-i)(i) ) (-6i)(i) 6) i(-i) 7) i( i) 8) 6i( + i) 9) -i( + 9i) 0) 6i( i) 1) (+i)( i) ) (-i)( + i) ) (6-7i)(6+i) ) (+i)( i) ) (+i) 6) (6 i) 7) (1 i) 8) ( i) 9) (6+i)(6 i) 0) (+i)( i) 1) (8+i)(8 i) ) (7 i)(7+i) ) (1+i)(1 i) ) ( i)(+i) 0
14 Chapter : Radicals and Complex Numbers Section.10: Complex Numbers # - 60: Simplify ) 1+ 9 (7) 6) () 7) 1 0 (1) 8) 8 98 () 9) + 0 () 1) (1) 0) 8 () ) + 11 () ) 6+ ()(1) (1) ) 6+ (9)(1) () ) 1 (1) 6) 8 () 7) 10 ( ) 9 () 9) (1) 8) 6 ( ) () 60) () 1
15 Chapter : Radicals and Complex Numbers Chapter : Review 1 have been deleted ) Evaluate the roots a) 11 b) 6 c) 9 : Create a table of values and sketch a graph of the function. You may use your calculator and should round to two decimal places when needed. Use the graph to find the domain and range of the function in interval notation. a) f(x) = x b) f(x) = x 6) Simplify the radical expression. (Assume all variables represent positive real numbers.) a) x 1 b) x8 6 c) 7x 1 y 6 7) Simplify the expression. a) 7 b) 7 c) ( 1 6 ) 8) Simplify the expression. Write your answer using only positive exponents. a) 1 b) x 1 x c) (y 1 9) Write the expression using radical notation. a) (x) 1 b) x 1/ y ) 10) Write the expression using rational exponents. a) x b) x 11) Multiply, then simplify. a) x 7x b) a b a 7 b 11 1) Simplify. a) 0 b) 0 c) 7xy 6 z 7 1) Add or subtract. a) 6 x x b) + 9 c)
16 Chapter : Radicals and Complex Numbers Chapter : Review 1) Multiply, then simplify. a) 6 b) xy x y c) (8 ab)( a b) d) ( ) e) ( + 6 )( ) #1-17: have been deleted #18-1: Simplify the expressions. 18) Problem 19 has been deleted 0) ) Subtract, write your answer in the form a+ bi (6-i) (-i) ) Multiply, write your answer in the form a+ bi (-i) ) Simplify 1+ 0 () # - 9: Solve the radical equation if possible. ) b = 6) x = 8 7) (x ) 1 = 8) x + = 1 9) x = x
17 Chapter : Radicals and Complex Numbers Grima MAT 10 Chapter Practice test 1) Evaluate the roots a) 11 b) c) 7 ) Simplify the expression. a) 6 b) 6 c) ( ) ) Has been deleted. ) Simplify the radical expression. (Assume all variables represent positive real numbers.) a) y b) 6x y 6 ) Simplify the expression. (y 1 y ) 6) Make a table of values and sketch a graph. State the domain and range of the function. a) f(x)= x b) f(x) = x 7) Write the expression using radical notation. a) (x) 1 b) x 1 8) Write the expression using rational exponents. a) x b) x 9) Multiply and simplify. a) 9x x b) x 1x 10) Simplify 7xy 7 11) Simplify the expressions. 98 1) Subtract, write your answer in the form a+ bi (--i) (-i) 1) Multiply, write your answer in the form a+ bi (1+i) 1) Multiply, then simplify. a 9 b 1) Simplify. 7xy 1 z 8a 7 b
18 Chapter : Radicals and Complex Numbers 16) Simplify the expression ) Multiply and simplify: ) Solve the radical equation: x = 19) Subtract, ) Multiply, then simplify. ( + 6 )( 6 ) 1) Simplify as much as possible () ) Solve the radical equation if possible. a) (x + ) 1 = 9 b) x = x c) x + 1 =
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