Lesson 2.2 Exercises, pages
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1 Lesson. Exercises, pages Write each mixed radical as an entire radical. a) 6 5 b) 6 # 5 # # 108 c) - 5 () # d) 5 5 # 5 8 # 5 65 # Write each entire radical as a mixed radical, if possible. a) 5 b) 9 # # c) d) # 7 7cannot be written as a mixed radical because 7 does not have any factors that are perfect fourth powers P DO NOT COPY.. Simplifying Radical Expressions Solutions 7
2 5. rrange in order from least to greatest. a) 6 7, 11 7, 5 7 b) 8, 5, Each mixed radical is a multiple of 7, so compare the coefficients: 5 < 6 < 11 So, from least to greatest: 5 7, 6 7,11 7 Each radical has index. Write each mixed radical as an entire radical. # 5 # 8 5 # 5 9 # 0 7 Compare the radicands: 0 < 7 < 8 So, from least to greatest: 5,, 8 B 6. For which values of the variable is each radical defined? Justify your answers. a) 18x b) -a 18x H when 18x» 0. a H when a» 0. 18> 0 and x» 0 < 0, so a 0; that is, a 0 So, 18x is defined for. So, a x H is defined for a 0. c) c d) 18z 5 Since the cube root of a number 18z 5 H when 18z 5» 0. is defined for all real numbers, 18 > 0, so z 5» 0; that is, z» 0 c when c So, 18z is defined for z» 0. H H. 5 So, c is defined for c H. 7. Write each mixed radical as an entire radical. a) 7 b) - 5 c) - 8 # 7 9a 7 b 1 a # 5 b 8 a 9 ba5 8 b Write each entire radical as a mixed radical, if possible. 15 a) b) c) a b 8 a # 7 ba b 5 # # 81 # 6 7 # 8. Simplifying Radical Expressions Solutions DO NOT COPY. P
3 9. rrange in order from greatest to least. a) 5, 11, 8 The mixed radicals have different indices, so use a calculator Á Á Á Compare the decimals: 6.771> 6.6> So, from greatest to least: 8, 11, 5 b) 6, 7,, 5 Each mixed radical has index and coefficient. So, compare the radicands: 7 > 6 > 5> So, from greatest to least: 7, 6, 5, c) 5, 1,, Each radical has index. Write each mixed radical as an entire radical. 5 # 1 # # # Compare the radicands: 15> 18> 108> 96 So, from greatest to least: 5,,, Write the values of the variable for which each radical is defined. Simplify the radical, if possible. a) b) 8b -8x Since the cube root of a 8b ç when 8b» 0. number is defined for all real 8> 0 and b» 0 values of x, the radical is So, 8b is defined for defined for x ç. b ç. 8b 8x 8 # 16 # x b # b x c) d) 16r 15x 16r ç when 16r» 0. 15x ç when 15x» 0. 16> 0 and r 15 0 so ; that is,» 0 > x» 0 x» 0 So, 16r is defined for r ç. So, 15x is defined for x» 0. 16r 16 # r 15x cannot be simplified because r the radicand does not have any factors that are perfect fourth powers. P DO NOT COPY.. Simplifying Radical Expressions Solutions 9
4 11. For which values of the variable is each radical defined? Rewrite the radical in simplest form. 7x -8c a) b) c) Since the cube root of a number Since the cube root of a number is defined for all real values of x, is defined for all real values of c, the radical is defined for x ç. the radical is defined for c ç. 8c 5 8c # 7x 6c 16 7 # x 8 # y y 5 x when y ; that is, 5» 0 y» 0 16 ç 16 81y # y 16 81y 5 y y 5 c 6c 1. Determine whether each statement is: always true sometimes true never true Justify your answer. a) 1-x = x lways true; a number multiplied by itself is always positive. b) x =;x Sometimes true; x x, and x» 0, so when x 0, x x. But, when x 0, x x 1. In the far north, ice roads are often the only way to get supplies to mines. The formula t = m is used to determine the minimum thickness of ice, t inches, needed for an ice road to support a mass of m tons. a) What is the minimum thickness of ice needed for a mass of 10 T? Use the formula t m. Substitute: m 10 t 10 So, the minimum thickness of ice needed is 10 in. 10. Simplifying Radical Expressions Solutions DO NOT COPY. P
5 b) What is the maximum mass for ice that is 10 in. thick? Use the formula t m. Substitute: t m a 5 b 5 m ( m) m 5, or 6.5 So, the maximum mass is 6.5 T. c) Suppose the mass of a load is multiplied by. How will the thickness of the ice have to increase? Explain your thinking. When the mass is multiplied by, t m # m Compare this to the original equation: t m The thickness of the ice will be multiplied by, which is. C 1. For which values of the variables is each radical defined? Simplify the radical, if possible. a) 98a b 6 b) -0x y 5 98a b 6 ç when 98a b 6» 0. Since the cube root of a number 98> 0 and b 6» 0 is defined for all real numbers, So, a is defined for» 0; that is, a» 0 0x x ç, y ç. y 5 is defined for 0x y 5 98a 8 # 5 # x y # b 6 y a» 0, b ç. 98a b 6 9 # # a b 6 # a 7ab a xy 5y c) 8r s 8 d) 18m n 8r s 8 ç when 8r s 8» 0. Since the cube root of a number 8> 0, r» 0, and s 8» 0 is defined for all real numbers, So, 8r s 8 is defined 18m is defined for m ç, n ç. n for r ç, s ç. 18m n 6 # # n # m n 8r s 8 16 # # r s 8 rs n m n P DO NOT COPY.. Simplifying Radical Expressions Solutions 11
6 15. The surface area of sphere B is square units. The surface area of sphere C is twice that of sphere B. The surface area of sphere D is 9 times that of sphere B. Determine the radii of spheres C and D in terms of. B D C Formula for surface area of a sphere: S r Sphere B has surface area square units. Sphere C Sphere D Surface area: Surface area: 9 So, r So, 9 r 9 r r 9 r r Sphere C has radius units. Sphere D has radius units. 16. circle with radius r and area touches another circle with radius y and area. larger circle is drawn around the other two as shown. a) Determine an expression for the value of y in terms of r. State the restrictions on the variables. r y Formula for area of a circle: r Since r and y are lengths, r>0 and y>0. rea of small circle: r rea of middle circle: y To determine y in terms of r, substitute r in y, then solve for y. y (r ) y r y r y y r y r b) Determine the area of the shaded region in terms of. rea of shaded region rea of large circle area of smaller circles (y r) Substitute: y r (r r) 5 (r) 5 16r 5 Substitute: r The area of the shaded region is Simplifying Radical Expressions Solutions DO NOT COPY. P
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