MATH 521B WORKBOOK RADICALS AND RATIONAL EXPRESSIONS
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1 MATH 5B WORKBOOK RADICALS AND RATIONAL EXPRESSIONS
2 Unit - Radicals Estimation of Radicals Estimate, then find an approximate value, to the nearest hundredth Estimate, then find an approximate value, to the nearest tenth Give a decimal approximation to the nearest hundredth fo each radical Use decimal approximations to arrange the following in order from least to greatest. 5. 7,, , 5, 7, , 7, 5, 6 Simplifying Radicals Simplify a) 6 b) c) 5 d) 8 e) f) 6 Simplify a) b) c) d) Simplify. Assume that each radical represents a real number.. 8x. x a c
3 7. Express the exact area of the triangle in simplest radical form. 8. A square has an area of 675 cm. Express the side length in simplest radical form. Write as an entire radical Without using a calculator, arrange the following in order from least to greatest. 5. 5,,, , 7, 8, , 7,, 0 Operations with Radicals a) add or subtract Simplify Express the volume of the rectangular prism in simplest radical form
4 Simplify. If no simplification is possible, say so Operations with Radicals b) Multiply and divide Simplify Expand and simplify.. ( 0 + ). ( 6 ) 5. 6( + 6) 6. ( 6 ) 7. ( + ) 8. ( 6 + 0) 9. ( 5 + 6)( 5 + 6) 0. ( )( + ). ( 7 )( ). ( + ). ( 5). ( + )( ) 5. ( 6 )( 6 + ) 6. ( 7 + 5)( 7 5) 7. Write and simplify an expression for the area of the square below. 8 5
5 triangle 8. Find the length of the hypotenuse in each isosceles right triangle. Write your answer as a mixed radical in simplest form. a) b) c) h h 5 h 5 9. What is the relationship between the length of the hypotenuse and the length of each leg in a right isosceles triangle? 0. Find the lengths of the indicated sides in each isosceles right triangle. a) b) c) d) 7 k h 0 a 6 c d h k a 8. If the length of each leg of a isosceles right triangle is, what is the length of the hypotenuse?. If the length of each leg of a isosceles right triangle is s, what is the length of the hypotenuse in terms of s? triangle. Find the lengths of the indicated sides in each triangle. Write radical answers as mixed radicals in simplest form. a) b) c) e) d b 8 k x 0 e a y g. What is the relationship between the length of the hypotenuse and the length of the side opposite the 0 angle? 5. What is the relationship between the length of the side opposite the 60 angle and the length of the 5
6 side opposite the 0 angle? 6. Find the lengths of the indicated sides in each triangles. a) b) c) k 8 a h h h 6 a 7. If the length of the hypotenuse is, what are the lengths of the other two sides? 8. If the length of the hypotenuse is s, what are the lengths of the other two sides in terms of s? The Equilateral triangle 9. Given equilateral triangle DEF, find a) the altitude, a b) the area, A 6
7 0. Given equilateral triangle XYZ, find a) the altitude, a b) the area, A. Given equilateral triangle PQR, express a) the altitude, a, in terms of s b) the area, A, in terms of s. Use your formula from question b) to find the area of each of the following triangles. a) b) c) Problem Solving. An equilateral triangle has an area of 6 cm. Calculate the side length and the height of the triangle, to the nearest tenth of a centimetre.. A regular hexagon has an area of 600 cm. Calculate the length of each side, to the nearest tenth of a centimetre. 5. A triangle has an area of 0 cm. Calculate the length of each side, to the nearest tenth of a centimetre. 7
8 6. The equilateral triangle is inscribed in a circle. The circle has an area of 6B. State the exact area of the triangle as a mixed radical in simplest form. 7. a) Find the area of triangle ABC, in square centimetres. b) Triangle DEF is similar to triangle ABC. Find the area of triangle DEF, in square centimetres. Simplify ( 7) 55. ( 6) Simplify. 8
9 58. ( 8 + 0) 59. ( ) 60. 5( + 5) 6. 7( ) 6. ( 8 5 ) 6. 5( ) 6. 5( 00 6) 65. 0( 5 + 5) Simplify. 66. ( + 7)( 7) 67. ( 5+ )( 5 ) 68. ( 7 + ) 69. ( 5 + ) 70. ( + )( + ) 7. ( 6 )( + ) 7. ( 7 ) 7. ( 0) 7. ( + )( ) 75. ( 5 )( ) 76. ( 7)( + 7) 77. ( )( + ) 78. ( 5+ )( 8 ) 79. ( + 6)( 5 6) 80. ( 5 + 7) 8. ( + 6) 8. ( + 5)( 5) 8. ( 7 5)( 7 + 5) 8. ( 6 5) 85. ( 5 0) ( )( 6 ) 88. ( 5 + 5)( 5) Rationalizing denominators Simplify Simplify
10 Simplify If a rectangle has an area of square units and a width of 7 5 units, what is its length in simplest radical form? 7. Express the ratio of the area of the larger circle to the area of the smaller circle in simplest radical form. + 0
11 Simplify a) b) c) d) Simplify. If no simplification is possible, say so
12 Rational Exponents Write in radical form x. a x 7 0. b 6 5. ( x). 6 x Write using exponents a ( b) b. x 5. 5t x 5 a ( x ) Evaluate ( ) ( 7) 6. ( 6) Evaluate ( ) 5 8. ( 7) ( ) ( ) 7 8 Simplify a) b) 50. a) b) a) b) 5. a) 6 b) 6
13 Simplify ( 5) ( 5 ) 65. ( ) 66. ( ) Express in simplest radical form Radical Equations Solve and check.. x = 5. x = 0. x + = 0. y + = 5. m = 6. x + = 0 7. x + = 8. x = 0 9. x + 6= 0. z + =. ( ) x ) =. x = 8. x + + = Solve. If an equation has no real solution, say so. 5. x = 5 6. n + = 7 7. t 5= a = 9. x 7 = y + = 9. m + =. w 5 =. d + 5 =. 7 9 c = Equations with Rational Exponents Give the power to which you would raise both sides of each equation in order to solve the equation.. x = 9. x =. x =. x = 8 Tell what steps you would use to solve each equation. 5. x = x = 0 7. ( x ) = 8. ( 5x) =
14 Solve each equation. a 9. a) = 8 b) ( x + ) = 8 0. a) y = 6 b) ( y) = 6. a) y = 0 b) ( y) = 0. a) ( 9t) = b) 9t =. ( 8 y) =. ( n ) = 5. ( + ) = 5 x 6. ( + 9) = 5 Exponential Equations x Solve.. =. 5 = 5. = = 6 Solve. If an equation has no solution, say so. 5. = x = x = x x x+ 5 x+ ( x ) = 0. 9 = 7 7. = 6. = 9 x x+ 6 x+ x x x. 5 = 5. 6 = = x
15 Answers for Unit - Radicals Estimation of Radicals. 9; ; ; ; ; ;.0 7. ; ; ,, 5. 5, 6, 7,8. 6, 7, 5, 5 Simplifying Radicals a) b) c) d) e) f) a) b) c) d). x. x x 5. 5 a a 6. c c
16 , 5,, , 7, 8, , 0, 5 5, Operations with Radicals a) add or subtract not possible. not possible
17 Operations with Radicals b) Multiply and divide a) b) c) 5 9. The length of the hypotenuse is times the length of a leg. 0. a) k = 7, h= 7 b) k = 0, h= 0 c) a = b=6 d) c= d =.. s. a) a =, b= b) x = 5, y = 5 c) d =, e= d) g =, k =. The length of the side opposite the 0 angle is half the length of the hypotenuse. 5. The length of the side opposite the 60 angle is times the length of the side opposite the 0 angle. 6. a) h=, a = 6 b) h= 6, k = 8 c) a =, h= 6 7., 8. s s, 9. a) b) 7
18 0. a) b) s. a) b) s. a) 9 b) 5 c) 9 Problem Solving. 9. cm, 7.9 cm. 5. cm 5..8 cm, 8. cm, 9.6 cm a) 6 cm b) cm Rationalizing denominators
19 a) + 8 b) a) b) c) d) Rational Exponents.. ( 7) 5. x. a 5. ( 6) 6. ( 6) x. x x 5 b ( ) 6. a b 9. x 0. a. x 5. b. x t 9
20 a) b) 50. a) b) 5. a) b) 5. a) b) ! Radical equations. 5..!. 5.! 6. No solution no solution No solution. 6.! no solution 9.!, 0.!,.. 6.!. 5 0
21 Equations with Rational Exponents a) 6 b) 5 0. a) b) a) b) a) b) 7 8.! ± 6. ± Exponential Equations ! No solution
22 Unit 5 - Rational expressions Rational Expressions. For which values of x are the following rational expressions not defined? x y x x x + x a) b) c) d) e) x y x + y x x 8 x f) x + 5 xy + y x 9y Find (a) the domain of each function and (b) its zeros, if any. t 9 x + x. f ()= t. gx ( ) =. F( x) = ( x 6)( x ) t 9t x t + t 9 s + 5s 5. hy ( ) = ( y 8)( y+ ) 6. gt ()= 7. Gs () = t t ( s ) x x + x t + t t 8. f ( x)= 9. ht ()= x + x t t + t Simplifying Rational Expressions Reduce to lowest terms. State any restrictions on the variables. x + y 6t 6 5x x + 5y t 6 x xy + 0 x y + y x x x + x
23 Express in simplest equivalent form. y + 0y + 5 r x y + 5 5r + 0 x y 6xy t 8t + 5x + xy y t t x + xy Simplify. a a y 8y + 5 n n... a 9a + 0 y 5 n + n 6 t t 6x x t t + 8x 6x 9 6. The area of a Saskatchewan flag can be represented by the polynomial x +x + and its width by x +. a) Write a rational expression that represents the length. b) Write the expression in simplest form. c) If x represents unit of length, what is the ratio length : width for a Saskatchewan flag? 7. The length of an edge of a cube is x +. Write and simplify a rational expression that represents the ratio of the volume to the surface area. 8. The diagrams show the numbers of asterisks in the first diagrams of two patterns. Pattern Pattern
24 9. (cont d.) a) For pattern, express the number of asterisks in the nth diagram in terms of n. b) For the pattern, the number of asterisks in the nth diagram is given by the binomial product (n + )(n + ), where the blanks represent whole numbers. Replace the blanks in the binomial product with their correct values. c) Divide your polynomial from part b) by your expression from part a). d) Use your results from part c) to calculate how many times more asterisks there are in the 0 th diagram of pattern than there are in the 0 th diagram of pattern. e) If a diagram in pattern has 0 asterisks, how many asterisks are in the corresponding diagram of pattern? f) If a diagram in pattern has 95 asterisks, how many asterisks are in the corresponding diagram of pattern? Simplify. 5x 5x u u 0... ( pq)( q p) 0x u + u s t x 5x + 6 6y 5y ( t s) x 7x + y 6y 6. ( r x + x 8 5r + )( r ) 7. ( x)( + x) x + x x x x y + xy y x x x + x y s t x + x y xy y 0.. s s t + t x y Multiplying or Dividing Rational Expressions Simplify. State any restrictions on the variables. 5 6ab x y 5xy 0xy... 6x y 9x y 8ab 6x y 9x y xy x 5( y ) y + ab 8ab x 6 y + 0 ( a + b) a + b x + 5x + 6 x x x 6x m m m x + 9x + 8 m + 5m m x + xy x 0xy + y a 9a x xy y x 9y a 9 7m+ + m5 a a
25 w 5w w + w6.. 8w + w 8w w5 n + 6 n + n + n.. 6x 7x + 6 x 9x + 8 x + 6x 5x x 9 x + x 5x x + 7x + 6x x 8x + x 5
26 Simplify. Write answers without negative or zero exponents. 5x t t 0x 8 9. rs 8s 9rs xx ( ) ( x ).. 5 7r 0 ( x ) x 5. x x x x 5x + x uv u uv v 8. u v u + v ( ) uv u v 9. x + xy y x + 7xy 6y x xy y x xy y p q ( p+ q) p + q x + y x + y x xy y 6 x u u u u 6
27 Adding or Subtracting Rational Expressions Simplify. In each of the following, state any restrictions on the variables. m+ m+ y 5 y. +.. t t + t
28 Simplify. In each of the following, state any restrictions on the variables x x + x x x x x x + y a a y 6 y + a 7a + a n 6n 5n+ m m.. + x + x + x m 9m+ 8 m m+ 0 y y 5. y 9 y y An RCMP patrol boat left Tofino on Vancouver Island and travelled for 5 km along the coast at a speed of s kilometres per hour. a) write an expression that represents the time taken, in hours. b) The boat returned to Tofino at a speed of s kilometres per hour. Write an expression that represents the time taken, in hours. c) Write and simplify an expression that represents the total time, in hours, the boat was at sea. d) If s represents 0 km/h, for how many hours was the boat at sea? 7. Simplify. In each of the following, state any restrictions on the variables. + x x t + t + + t y + y n n n 9 a) b) c) d) e) f) m m ( 5x + ) x ( 8x + ) x 8. Write two rational expressions with binomial denominators and with each of the following sums. Compare your answers with a classmates s. 5x + 8 5x 5 x x a) b) c) d) ( x + )( x + ) 6x x + 6 ( x )( x ) x 9 8
29 Simplify. 5 5 t x + x s + s + s. 5. x 5x x t t + + t u uv + v u v x xy + 9y xy y t + 6 Rational Equations Solve each open sentence. x t t + z z. + =. = = xx ( + ) x+ y( y ) y( y + ). = 5. + = Pump A can unload a ship in 0 h and pump B can unload the ship in h. Because of an approaching storm, both pumps were used. How long did both pumps take to unload the ship? 7. An old conveyor belt takes h to move one day s coal output from a mine to a rail line. A new belt can do the same job in 5 h. How long does it take when both belts are used at the same time? 8. A river boat paddled upstream at km/h, stopped for h of sightseeing, and paddled back at 8 km/h. How far upstream did the boat travel if the total time for the trip, including the stop, was 7 h? Solve and check. If an equation has no solution, say so = 0. y y + y 6. = t t. x 6. x x + x + x. x + = x x x u 0 + = 9 u u+ 5 u u + u 6 = u 9
30 5. A town s old street sweeping machine can clean the streets in 60 h. The old sweeper together with a new sweeper can clean the streets in 5 h. How long would it take the new sweeping machine to do the job alone? 6. The intake pipe can fill a certain tank in 6 h when the outlet pipe is closed, but with the outlet pipe open it takes 9 h to fill the tank. How long would it take the outlet pipe to empty a ful tank? 7. An excursion boat travels 5 km upstream and then back again in h 8 min. If the speed of the boat in still water is 5 km/h, what is the speed of the current? 8. Members of a ski club contributed equally to obtain $800 for a holiday trip. When 6 members found that they could not go, their contributions were refunded and each remaining member then had to pay $0 more to raise the $800. How many went on the trip? 0
31 Answers for Unit 5 - Rational expressions y y. a) x = y b) x = c) x = 0 d) x = e) x = ± f) x =±. Reals except 0 and 9; ±. Reals except ±; 0. Reals except ; ± 5. Reals except!; 6. Reals except 0,±;!, 7. Reals except ;, 8. Reals except ±; 9. Reals except ; ±,! Simplifying Rational Expressions 5, x y 6,t 6 5, x 5x 0 y, y. y + 5, y 5 r, r t 9. t 0 5x y a +,, 0., x 0, y., a 5, t x a 5 y n + t +., y ± 5., n,., t, y + 5 n + t 5. x, x, x + x + x + 6.a) b) x + c) : x + ( x + ) x + 7. = 6( x + ) 6 8. a) n + b) ( n+ )( n+ ) c) n + d) e) 0 f) 5 x u s+ t x x u s t x y r y r x + s + t x + xy + y x x + y ( s+ t)( st) x + y x, x 0, x + x +, x 0,, y 0 xy
32 Multiplying or Dividing Rational Expressions xy. 0.. ab, abxy,,, x, xy, 0 9 x y, x, y 0 y a., x 5., y 6., 6b a bab,, 0 x + m + 7., x 6, 5,, x m m x, 50,,,, x y x 6 + y, ± 6 y, 7 y a 5 ( w+ )( w+ 5) 0., a, ±. a + ( 7)( ), 7, 5,, w w+ w+. ( x + 5 )( x ), x,,,, ( x + ) 0. ( n ) 5x +, n, ±., x,,, x , 0x x 5. a) b) c) d) no, the answer is c) is independent of x ( x )( x ) 7. y + ( 6x 9)( x + ) ( x )( x + 6) 8. a) b) c) x 9. ( x ) r 0. x. t. y.. u + x + p q u x + y u + x p+ q v u x + y Adding or Subtracting Rational Expressions x ( x )( x ) 0m + 9 y t +... x +. a) b) 58,, ( x + )( x ) ( x )( x ) 5. a) b) c) 6 ( x )( x ) 6
33 x ( x )( x ) d) e) 6 nn ( + ) 6. a) b) 5,, 8, 6, 5 c) square d) e) ( n + ) f) It is a square. ( n+ )( n+ ) 5x x ( x + )( x + ),, 8x x 0 xx ( ),, x x x ± ( x 0 )( x + ), y y ±. ( y )( y + ), 9a a a ( a )( a ),, 8n 9. n. ( n)( n),, x 8 x ± ( x + ) ( x ), 6m 6m. m ( m)( m6)( m5),,, y 5y y ± ( y ) ( y + ), a) b) c) d) 6.75 h s s s x + 7. a) 0 y +, x, b) y x ( y ), m + c) d) m m t, 0,, t 0, t + n + e) n ± f) ( n ), ( x + ) ( ), x x +, 8. Answers may vary. 9. t ( s+ ) ( s) 5x t. 5. ( x )( x + )( x ) ( t ) ( t + ) x x x y( x y)
34 Rational Equations , 5,, h 8. 6 km h 9. No solution 0..,.,. 0,.! 5. 0 h 6. 8 h 7..5 km/h 8. 0
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