 # Test on Topic 16 Radicals Solutions

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Test on Topic 16 Radicals Solutions
Math 99 – Topic 16 Test on Radicals Test on Topic 16 Radicals Solutions 1. Evaluate each radical expression, if possible (If it is not a real number, so state.): 36 81 3  64 (a) 49 (b) (c) –  4 (d) 1 16  1 (e) 4 (f) 4 16 = 2 (g) –  25 5 (h) 2. Evaluate each exponential expression, if possible (If it is not a real number, so state.): 11 4 3 3 (8) 2 3  1 (a) (b) 25 2 (c) 3 3. Express each of the following in their simplest exponential form. x  (a) 3 27 x15 (b) x 7 (c) 5 2 4 5 4. Simplify the following expressions giving your answer in radical form.  x   4 y  6 4 y13 (a) x 6 (b) 3 y 17 (c) 5 (d) (e) 5 3  16z  6  2 1 5. Simplify (write the answer in radical form without a negative exponent): 6. Simplify (Write exact expressions using radicals, not decimals from a calculator). Assume the variables represent positive numbers. 3 49 6 5 (a) (e) 80 64 81 (b) (f) 20 25 4 (c) (g) (d) (h) 3 32 12 5 Page | 1

Math 99 – Topic 16 Test on Radicals
7. Simplify (Write exact expressions using radicals, not decimals from a calculator). (a) (d) (g) 3 27 x3 y10 z 5 8x 3 y 3 z 5  2x 2 y 3 z 64 xy 8x 7 y 5 (b) (e) (h) 2x 4 y 5 z 6  8xy 6 z 64 xy 8x 7 y 5 2 y 3 z 3 (c) (f) (i) 3 40x 6 y 5 z 19 72x5 y 4 z 3 36x 3 y 5 w4 8 yz 8. Simplify by rationalizing the denominator. 12 3 8 y 2 z 5 2 y 36 49 (a) (b) (c) 10 x 2 y 40 xy 5 8 2 (d) 8x y z 3 4 5 (e) (f) 3x 2 y 3 4 xy 4 81 4 81 2 9 10 x 3 y 5 2 xy (g) (h) (i) 9. Simplify the following 5  2 5  2  20  3 2  2 5 (a) 12 + 27 (b) 8 + (c) 10. Simplify (Write exact expressions using radicals, not decimals from a calculator). Assume the variables represent positive numbers. 64 81 8x 3 y 4 z 5 12 3 3 49 36 3x 2 y 3 4 xy 6 5 10 x 2 y 40 xy 5 8 y 2 z 5 2 y 12 5 8 2 10 (a) (e) (i) (b) (f) (j) (c) (g) (k) (d) (h) (l) Page | 2

Math 99 – Topic 16 Test on Radicals
11. Solve the following Radical Equations. (a) x = 9 (b) 4x  3 = 9 (c) 2 x = 8 (d) 3x  6 = 3 12. Solve the following Radical Equation. 2x  3 = x 13. Solve the following Radical Equations 2x  17 = x  7 14. Solve the following Radical Equations. (a) 4x  5 = 5 (b) x  2  7 = 1 15. Solve the following Radical Equation 2x  17 = x + 1. Page | 3

Test on Topic 13 Radicals Solutions
Math 99 – Topic 16 Test on Radicals Test on Topic 13 Radicals Solutions 1. Evaluate each radical expression, if possible (If it is not a real number, so state.): (a) 49 = 7 or – 7 36 81 36 81 6 9 2 3 (b) = = = (c) –  4 = Not real number 3  64 (d) = – 4 1 16 4 1 1 2 1 2 (e) 4 = = or – 16 (f) 4 16 = 2 (g) –  25 = Not a real number 5  1 (h) = – 1 2. Evaluate each exponential expression, if possible (If it is not a real number, so state.): (8) 2 3 (a) = 3 (8) 2 = 3 64 = 4  1 1 25 1 25 1 5 (b) 25 2 = = = 1 2 11 4 3 3 11  3 8 (c) = 3 4 4 = 3 4 = = 9 3 3. Express each of the following in their simplest exponential form. 15 (a) 3 27 x 15 = 3 27  x 3 15 = 3 x 3 = 3x5 7 (b) x 7 = x 2 x  5 2 . 4 5 10 20 1 x 2 5 2 (c) 4 5 = x = x = Page | 4

 x   4 y     16z  16 z   16z  1 1 1 4  5 = y13 y 3 4 y 64
Math 99 – Topic 16 Test on Radicals 4. Simplify the following expressions giving your answer in radical form. (a) x 6 = x3 (b) 3 y 17 = y 5 3 y 2  x   52  6 6  x  (c) 5 =   = x15 4 y13 y y (d) =  4 y    5 3 (e) 43 y5 3 64 y15 = 64  y15 = 8 y y = =  16z  6  2 1 5. Simplify (write the answer in radical form without a negative exponent):  16z  6  2 1 1 6 2 1 16 z 6 1 16 z 6 1 4 z 3 = = = = 16 z  6. Simplify (Write exact expressions using radicals, not decimals from a calculator). Assume the variables represent positive numbers. 80 16  5 = 16  5 (a) = = 4 5 (b) (c) 20 3 49 = 4  5 3 49 = 4  5 = 3 7 2 5 (d) 3 32 = 3 8  4 = 3 8  3 4 = 23 4 64 81 25 4 64 81 25 4 8 9 5 2 (e) (f) = = 6 5 6  5 5  5 30 5 (f) = = 12 5 12 5 12  5 5  5 60 5 4 15 5 4  15 5 2 15 5 (g) = = = = = = Page | 5

Math 99 – Topic 16 Test on Radicals
7. Simplify (Write exact expressions using radicals, not decimals from a calculator). (b) 3 27 x3 y10 z 5 = 3 27  3 x3  3 y10  3 z 5 = 3xy3 y z 3 z 2 3 = 3xy3z 3 yz 2 (c) 2x 4 y 5 z 6  8xy 6 z = = 2x 4 y 5 z 6  8xy 6 z 16x5 y11z 7 16  x5  y11  z 7 4x2 x y5 y z3 z 4x2y5z3 xyz (c) 3 40x 6 y 5 z 19 = 3 40  3 x 6  3 y 5  3 z 19 = 3 8  5  3 y 5  3 z 19 23 5  x2  y  z 6  3 z 2x2 yz z = (d) 8x 3 y 3 z 5  2x 2 y 3 z = 16x 5 y 6 z 6 = 16 x 5 y z 6 = 4x2 x y3z3 = 4x2y3z3 x 64 xy 8x 7 y 5 8 6 8 x6  y 4 2 2 x3 y 2 (e) = = = x y 4 72x5 y 4 z 3 72  x5  y 4  z3 6 2  x2 x  y 2  z z = 6x2 y xz (f) = = 64 xy 8x 7 y 5 8 6 8 x6  y 4 2 2 x3 y 2 (g) = = = x y 4 2 y 3 z 3 2 y 3 z 8 yz 3 y 2 4 z 2 y 2 4  z y 2 z (h) = = = = 8 yz 2 (i) 36x3 y5 w4  36  x3  y5  w4  6  x x  y y  w2  6xy 2 w2 xy Page | 6

Math 99 – Topic 16 Test on Radicals
8. Simplify by rationalizing the denominator. 12 3 12 3 3 12 3 3 (a) = = = 4 3 8 y 2 z 5 2 y 8 y 2 z 5 2 y 2 y 8 y 2 z y 2 y (b) = = = 4 yz y 36 49 36 49 6 7 (c) = = (d) 8x 3 y 4 z 5 = 8 x 3 y z 5 = 2 2 x x y 2 z z = 2xy 2 z 2xz 10 x 2 y 40 xy 5 x 4 y 2 x 4 y 2 x 2 y (e) = = = 8 2 8 2 2 8 2 2 (f) = = = 4 2 3x 2 y 3 4 xy 3x 2 y 3 4 xy 4 xy 3 x 2 y xy 4 xy 3 xy x y 4 6 xy xy 4 3 xy xy 2 (g) = = = = = 4 81 (h) 4 81 2 9 10 x3 y 5 2 xy 10 x3 y5 2 xy 2 xy 10 x3 y xy 2 xy 5 x 2 y xy 1 (i)  5x2 y xy 9. Simplify the following (a) 12 + 27 = 2 3  3 3  5 3 (b) 8 + 20  3 2  2 5  2 2  2 5  3 2  2 5   2 5  2 5  2   25  5 (c) 2  5 2  2  3 Page | 7

Math 99 – Topic 16 Test on Radicals
10. Simplify (Write exact expressions using radicals, not decimals from a calculator). Assume the variables represent positive numbers. 3 49 64 81 3 49 64 81 3 7 8 9 (a) (b) = = 6 5 6  5 5  5 30 5 (c) = = 12 5 12 5 12  5 5  5 60 5 4 15 5 4  15 5 2 15 5 (d) = = = = = = 36 49 36 49 6 7 (e) = = (f) 8x 3 y 4 z 5 = 8 x 3 y z 5 = 2 2 x x y 2 z z = 2xy 2 z 2xz 10 x 2 y 40 xy 5 x 4 y 2 x 4 y 2 x 2 y (g) = = = 8 2 8 2 2 8 2 2 (h) = = = 4 2 3x 2 y 3 4 xy 3x 2 y 3 4 xy 4 xy 3x 2 y xy 4 xy 3xy x y 4 6 xy xy 4 3xy xy 2 (i) = = = = = 12 3 12 3 3 12 3 3 (j) = = = 4 3 8 y 2 z 5 2 y 8 y 2 z 5 2 y 8 y 2 z y 2 y (k) = 2 y = = 4 yz y 5 10 5 10 10 5 10 10 10 2 (l) = = = 11. Solve the following Radical Equations. (a) x = 4x  3 = 9 (b) 9 x = 81 4x – 3 = 81 4x x = 84 21 Page | 8

Math 99 – Topic 16 Test on Radicals
2 x x = 8 4 16 (d) 3x  6 = 3 3x – 6 3x = = 9 15 x 12. Solve the following Radical Equation. = 5 2x  3 2x + 3 = x x2 x2 – 2x – 3 (x – 3)(x + 1) x = 3 or x = – 1 but after checking x = – 1 is unusable its a “phantom solution” so only x = 3 works 13. Solve the following Radical Equations. 2x  17 2x + 17 2x x x  7 x – 7 x – 24 – 24 = (after check x = – 24 is not a usable solution so there is no solution to this equation) 14. Solve the following Radical Equations. 4x  5 = (b) x  2  7 (a) 5 = 1 2 3x  2 4x + 5 = 25 = 8 3x  2 4x = 20 = 4 x = 5 3x + 2 = 16 3x x = 14 15. Solve the following Radical Equation 2x  17 = x + 1. 2x  17 2x + 17 17 = x+1 (x +1)2 (x + 1)(x + 1) x2 + 2x + 1 x2 + 1 x2 – 16 (x + 4)(x – 4) So x = – 4 or x = 4 but after checking x = – 4 is unusable so only x = 4 is a solution Page | 9