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`, , MATHS, BOOKS - RD SHARMA MATHS (HINGLISH), EXPONENTS OF REAL NUMBER, , 1. Evaluate each of the following 52, , ×, , 5, , 4, , (ii) (3 2 ), , 3, , Watch Video Solution, , 2. Evaluate each of the following (, , 3, , 11, , ), 12, , (ii) (, , 3, , −3, , ), , 4, , Watch Video Solution, , 3., , Evaluate, 2, , (i) (, , 4, , ), 11, , each, 11, , ×, , (, , 2, , ), 3, , of, 3, , ×, , (, , 3, , ), 2, , the, , following:

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5, , 1, (i) (, , 4, , −2, , ), , ×, , (, , 2, , ), , −1, , 3, ×, , 3, , (, , ), 5, , Watch Video Solution, , 4. Evaluate each of the following:, 2, (ii) (, , 2, , −3, , 2, , ), , ×, , (, , 3, , ), , 3, ×, , 5, , (, , (i) 2, , 55, , ×, , 2, , 60, , −, , 2, , 97, , ×, , 2, , 18, , 2, , ), 5, , Watch Video Solution, , 5. If, a, , a, , a = 2, b, , + b, , and, , (b) ab, , , then find the values of each of the following:, , b = 3, , +, , a, , b, , (c) ab, , Watch Video Solution, , 6. If, a, (, , a = 2, , b = 3, , 1, , 1, , a, , ), b, , and, , (b) (, , +, a, , , then find the values of each of the following:, a, , ), b, , Watch Video Solution

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7, , 7. Simplify the following (3a, , 4, , 3, , 3, , 5, , b )(18a b ), , 6, , (ii), , 6, , 3, , 2, , 3a b, , − 2a, (, , 8, , ), 3, , 18a b, , b, , Watch Video Solution, , 8., , Simplify, 5, , n, , − 6, , (b), 9, , ×, , 5, , n, , each, ×, − 2, , 5, 2, , of, , the, , following:, , n+1, , ×, , 16, , 5, , n, , 7, , ×, , 2, , (c), 16, , ×, , 2, , n, , − 3, , (a), n+1, , n+2, , 20, , ×, , 7, , − 4, , ×, , 2, , − 2, , ×, , 2, , n, , ×, , 7, , n+1, , − 2, , ×, , 7, , n, , n, , n+2, , Watch Video Solution, , 9, , 9., , n, , ×, , 3, , 2, , ×, , If, 3, , (3, , 3m, , −, , ×, , m − n = 1, , Watch Video Solution, , −2, , n, 2, , ), , 2, , 3, , −, , 27, , n, , 1, =, 27, , ,, , prove, , that

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10. Assuming that, , x, , is a positive real number and, , numbers, show that: (, , x, x, , a, , b, , c, , ), , x, (, x, , b, , c, , a, , ), , x, , ), x, , are rational, , c, , a, , (, , a, b, c, , = 1, , b, , Watch Video Solution, , 11. Assuming that, , x, , is a positive real number and, 1, , 1, , numbers, show that: (, , x, , a, , ), x, , x, , ab, , (, , b, , x, , b, , c, , are rational, , 1, , bc, , ), , a, b, c, , x, (, x, , c, , a, , ac, , ), , = 1, , Watch Video Solution, , 12. Assuming that, , x, , is a positive real number and, , numbers,, x, , a, , (, x, , show, , a, , ), b, , a, b, c, , 2, , x, , ^ 2 + ab + b (, x, , Watch Video Solution, , b, , c, , that:, , b, , ), , are rational, , 2, , x, , ^ 2 + bc + c (, x, , c, , a, , c, , ), , 2, , ^ 2 + ca + a

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13. Assuming that, , is a positive real number and, , x, , numbers, show that: (, , x, , a, , x, , b, , a+b, , ), , x, , b, , x, , c, , (, , b +c, , ), , x, (, x, , are rational, , c +a, , c, , a, , a, b, c, , ), , = 1, , Watch Video Solution, , 14. If, , x, , is a positive real number and the exponents are rational, , numbers, show that: (, , x, , a, , a+b −c, , ), x, , x, (, , b, , x, , b +c −a, , b, , c, , ), , x, (, x, , c +a−b, , c, , a, , ), , = 1, , Watch Video Solution, , 15., , Show, , x, , that:, , (i), , a+b, , ) (x, , b +c, , ), , 2, , (ii), (x, , a, , b, , c, , (x, , c +a, , 4, , x x ), , Watch Video Solution, , 2, , ), , = 1, , x, ÷ (, , x, (x, , a(b −c ), , b (a−c ), , x, , b, , a, , c, , ), , = 1,

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16., , Show, 1, , 1 + x, , b −a, , that:, , 1, + x, , +, , c −a, , 1 + x, , a−b, , 1, +, , x, , c −b, , +, 1 + x, , b −c, , +, , x, , a−c, , = 1, , Watch Video Solution, , 17., , If, , show, , abc = 1,, , 1, , 1, , 1 + a + b, , −1, , that, , 1, , +, 1 + b + c, , −1, , +, 1 + c + a, , −1, , = 1, , Watch Video Solution, , a, , 18. Prove that:, a, , −1, , −1, , + b, , a, −1, , +, a, , −1, , −1, , 2, , 2b, , − b, , −1, , =, , 2, , b, , 2, , − a, , Watch Video Solution, , 19. If, 2, , 3, , 4, , a b c, , a, b, c, , are distinct positive prime integers such that, , = 49392,, , find the value of a,, , Watch Video Solution, , b, , and c

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20. Find the value of x, if 5x − 3, , x 3, , 2x − 8, , = 225, , Watch Video Solution, , 21. Solve the following equations: 2x − 5, , = 256, , (ii) 2 x + 3, , Watch Video Solution, , 22. If 9x + 2, , = 720 + 9, , x, , , find the value of (4x), , 1, x, , Watch Video Solution, , 23. Solve the equation for x :, , 2, , 2x + 1, , = 17. 2, , x, , − 2, , Watch Video Solution, , 24. Solve the equation for x 52x + 1, , = 6. 5, , x, , − 1, , 3, , = 4, , x −1

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Watch Video Solution, , 25. 3(a4 b3 ), , 10, , 2, , 2, , 3, , × 5(a b ), , Watch Video Solution, , 26. Simplify that: (2x − 2 y 3 ), , 3, , Watch Video Solution, , 27. Simplify that:, , (4 ×, , 7, , 10 )(6, 8, , ×, , ×, , 10, , Watch Video Solution, , 2, , 28. Simplify that:, , 4ab, , 3, , ( − 5ab ), 2, , 2, , 10a b, , Watch Video Solution, , 4, , 10, , −5, , )

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29. Simplify that : (, , x, , 2, , y, , 2, , n, , 2, , ), , 3, , a b, , Watch Video Solution, , 2n − 9, , 30. Simplify that:, , (a, , ), , 6, , 2n − 4, , a, , Watch Video Solution, , 31. If, b, , (ii) a, , a = 3, a, , + b, , and b =, , − 2 ,, , find the values of:, , a, , (i) a, , ab, , (iii) (a + b), , Watch Video Solution, , 32. Prove that: (, , x, , a, , 2, 2 + ab + b, , a, , ), x, , b, , Watch Video Solution, , x, ×, , (, x, , b, , c, , b, , ), , 2, 2 + bc + c, , x, ×, , (, x, , c, , a, , c, , ), , b, , + b, , 2, 2 + ca + a

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33. Prove that: (, , x, , c, , a, , ), x, , x, ×, , (, , b, , x, , a, , b, , c, , x, , ), , ×, , (, x, , b, , c, , a, , ), , = 1, , Watch Video Solution, , 1, , 34. Prove that:, , 1, +, , 1 + x, , a−b, , = 1, 1 + x, , b −a, , Watch Video Solution, , 35., , Prove, 1, , 1 + x, , b −a, , that:, , 1, +, , x, , c −a, , +, 1 + x, , a−b, , 1, +, , x, , c −b, , +, 1 + x, , b −c, , Watch Video Solution, , a + b + c, , 36. Prove that:, a, , −1, , b, , −1, , + b, , Watch Video Solution, , −1, , c, , −1, , + c, , −1, , a, , −1, , = abc, , + x, , a−c, , = 1

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37. Prove that: (a − 1, , + b, , −1, , ab, , −1, , ), , =, , a + b, , Watch Video Solution, , 38., , If, 1, , 1, , 1 + a + b, , −1, , show, , abc = 1,, , that, , 1, , +, , +, 1 + b + c, , 1 + c + a, , = 1, , −1, , Watch Video Solution, , 39., , Simplify, , the, , 3, , following:, , (i), 3, , 5, , ×, , 25, , n+1, , −, , 25, , ×, , (ii), 5, , ×, , 5, , 2n + 3, , −, , 5, , n, , ×, , n−1, , 9, , ×, , n+1, , 9, , n−1, , ,, , 2n, , n+1, , (25), , Watch Video Solution, , 40., , Simplify, , the, , following:, , 5, , n+1, , +, , 3n − 2, , 16(2), , (ii), 3n + 1, , 10(2), , n, , − 7 (8), , −, , (i), 9, , 6(8), , n+3, , ×, , 5, , n, , 6, , ×, , − 2, , 2, , 5, , n+1, , ×, , 5, , n

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Watch Video Solution, , 41., , Solve, , (ii) 2, , x +1, , the, , = 4, , x −3, , following, (iii) 2, , equations, , 5x + 3, , = 8, , for, , x :, , (i) 7, , 2x + 3, , = 1, , x +3, , Watch Video Solution, , 42., , Solve, , (ii) 4, , x −1, , the, ×, , following, 3 − 2x, , (0. 5), , equations, 1, , = (, , for, , x :, , (i) 4, , 2x, , 1, =, 32, , x, , ), , (iii) 2, , 3x − 7, , = 256, , 8, , Watch Video Solution, , 43. Solve the following equations for, (ii) 3, , 2x + 4, , + 1 = 2. 3, , x +2, , Watch Video Solution, , x : (i) 2, , 2x, , −, , 2, , x +3, , + 2, , 4, , = 0

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44. If, , 4, , 2, , 3, , 49392 = a b c ,, , a, b and c, , find the value of, , a, b and c, , , where, , are different positive primes., , Watch Video Solution, , 45. If, , 1176 = 2, , a, , ×, , 3, , b, , ×, , c, , 7 ,, , and, , a, b and c, , are natural numbers., , find a,b,c., Watch Video Solution, , 46. Given, , a, , b, , c, , 4725 = 3 5 7 ,, , find the integral values of, , a, b and c, , the, , value of 2 − a 3b 7c, Watch Video Solution, , 47. If, q −r, , a, , a = xy, , r−p, , b, , p −q, , c, , p −1, , , b = xy, , = 1, , Watch Video Solution, , q −1, , and, , c = xy, , r−1, , , prove that

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1, , 48. Find the value of: (, , 3, , 8, 27, , ), , Watch Video Solution, , 1, , 49. Find the value of : (, , 32, , 5, , ), 243, , Watch Video Solution, , 1, , 50. Find the value of: (, , 6, , 1, ), 64, , Watch Video Solution, , 51. Evaluate each of the following by removing radical signs and, negative indices wherever they occur:, (iii) (27), , −, , 2, 3, , (iv) (, , −, , 64, , 3, 2, , ), 25, , Watch Video Solution, , 1, , (i) (64) 3, , (ii), , (125), , −, , 1, 3

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52. Evaluate each of the following by removing radical signs and, negative indices wherever they occur: (27), , −, , 2, , (ii) (, , 3, , −, , 64, , 3, 2, , ), 25, , Watch Video Solution, , 5, , 53. Simplify each of the following: (i) (625), , −, , 256, , 1, 4, , (ii) (, , 4, , ), 81, , Watch Video Solution, , 54. Simplify each of the following: (i) (, , ), 32, , Watch Video Solution, , 55. [{(625), , −, , −, , 1, 4, , }, , 1, , 2, , 2, , ], , −, , 243, , 4, 5, , (ii) (32), , −, , 3, 5

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Watch Video Solution, , (, , −3, , − (4, , 2, , ), , ), , 56. Simplify: (256), Watch Video Solution, , 4, , 57. Simplify:, , 1, , (216), , −, , 2, , 2, , +, (256), , 3, , −, , 3, , +, (243), , 4, , Watch Video Solution, , 58. Simplify: √, , 1, + (0. 01), 4, , Watch Video Solution, , 3, , 59. Simplify :, , 3, , (25) 2 x(243) 5, 5, , 4, , (16) 4 x(8) 3, , Watch Video Solution, , −, , 1, , 2, , 2, , − (27) 3, , −, , 1, 5

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60. Simplify: (, , −, , 81, ), , 3, , −, , 25, , 4, , x[(, , 16, , 3, , −3, , 5, , 2, , ), , ÷ (, , 9, , ), , ], , 2, , Watch Video Solution, , 61. Simplify each of the following by removing radical signs and, negative, , indices, , (ii) (√5), , −3, , ×, , wherever, , (√2), , −3, , they, , −, , occur:, , (i) (√4), , 3, 4, , 1, (iii), 4, , −5, , 3, , Watch Video Solution, , 62. Simplify each of the following by removing radical signs and, negative, (ii) (81), , indices, −, , wherever, , 1, 2, , they, , occur:, , −7, , (iii) (√4), , Watch Video Solution, , −5, , ×, , (√2), , (i) (25), , −, , 1, 2, , ×, , 2, , √16

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63. Assuming that, , x, y, z, , are positive real numbers, simplify each of, , the following: √x − 2 y 3 (ii) (x −, , 2, 3, , y, , 2, , 1, , −, , 2, , ), , Watch Video Solution, , 64. Assuming that, , x, y, z, , are positive real numbers, simplify each of, 5, , the following: (i) (√x − 3 ) (ii) (√x), , −, , 2, 3, , √y, , 4, , ÷, , √xy, , −, , 1, 2, , Watch Video Solution, , 65. Assuming that, , x, y, z, , are positive real numbers, simplify each of, 4, , the following: (i)√xy 2, , 3, , 2, 2, ÷ x y (ii) √√x, , 3, , Watch Video Solution, , 66., , If, , x, y, z, ., , √x, , −1, , y√y, , −1, , are, , positive, , ., , z√z, , −1, , x = 1, , real, , numbers, , show, , that:

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Watch Video Solution, , 67., , x, , If, , (, , −1, , 3, , x y, a + b =, , − 1, , y, , 1, , 2, , −2, , 1, , x, , 6, , ) . (, x, , , where x, , and y, , y, 2, , −3, , y, , 3, , 2, , a, , ), , b, , = x y ,, , prove, , that, , are different positive primes., , Watch Video Solution, , 68., , 1, , If, , x =, , 28, , 3, , 1, , and y =, , 27, , find, , 3, , the, , value, , 1, x − y −, x, , 2, , + xy + y, , 2, , Watch Video Solution, , 69. If x, , = 3,, , find the value of (x, , 1, 3, , +, , x, , −, , 1, , 2, , 3, , ) (x 3 + x, , Watch Video Solution, , 70. If 25n − 1, , + 100 = 5, , 2n − 1, , , find the value of n :, , −, , 2, 3, , − 1 ), , of

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Watch Video Solution, , 71., , Solve, , the, , equation, , for, , x:, , (i) 4, , 2x, , 1, =, 32, , (ii) √(, , 3, , 1 − 2x, , ), 5, , 17, = 4, 27, , Watch Video Solution, , 72. Solve the equation for x :, , (5, , 0, , 2, , 2, +, , 3 − 2x, , ) 3 = (0. 6), 3, , Watch Video Solution, , 73. Solve the equation for x :, , 3, , 2 (5, , 0, , + 3, , 2x, , 8, ) =, , 8, , 27, , Watch Video Solution, , 74. Solve the equation: 3(2x, , + 1) − 2, , x +2, , + 5 = 0

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Watch Video Solution, , 75. Solve the equations for x, , and y 3, , x, , = 9, , ×, , y, , 3 , 8, , ×, , 2, , y, , = 4, , x, , Watch Video Solution, , 76., , Solve, x, , (32), , ÷ 2, , y +1, , the, , = 1, 16, , 4−, , equation, x, , − 8, , 2, , y, , for, , = 0, , Watch Video Solution, , 77. If ax, , y, , = b, b, , z, , = c and c, , = a,, , prove that xyz, , = 1, , Watch Video Solution, , 78. If ax, , y, , = b, , z, , = c, , 2, , and b, , = ac,, , prove that y, , 2xz, =, x + z, , Watch Video Solution, , x and y

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79. Assuming that, , are positive real numbers, simplify each of, , x, y, z, , 5, , 2, , the following: (i) (√x − 3 ) (ii) √x − 2 y 3 (iii) (x −, , 3, , y, , −, , 2, , 1, 2, , ), , Watch Video Solution, , 80. Assuming that, , x, y, z, , the following: (√x), , −, , are positive real numbers, simplify each of, , 2, 3, , √y, , 4, , ÷, , √xy, , −, , 1, 2, , (ii) √243, 5, , x, , 10, , 5, , y z, , 10, , Watch Video Solution, , 81. Assuming that, , x, y, z, , are positive real numbers, simplify each of, 5, , 5, , the following: (i) (, , x, y, , −4, , − 10, , √2, , 4, , ), , (ii) (, , 6, ), , √3, , (, , 2, , ), 7, , Watch Video Solution, , 5, , 82. Simplify: (i) (16, , −, , 1, 5, , 2, , ), , (ii) (32), , −, , 3, 5, , (iii) (343), , −, , 2, 3

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Watch Video Solution, , 3, , 83. Simplify: (i) (0. 001), , 3, , (25) 2, , 1, , (ii), , 3, , ×, , (243) 5, , 5, , 4, , (16) 4, , ×, , (8) 3, , Watch Video Solution, , 8, , 84., , Simplify:, , (i) (, , √2, ), , ÷ (, , 5, 7, , 5, (ii) (, 5, , 2, , −1, , ×, , ×, , 7, 7, , 2, , 2, , −4, , ), , 13, , √2, , 5, ×, , (, 5, , −2, , 3, , ×, , ×, , 7, , 7, , −, , 3, , −5, , ), 5, , 5, 2, , ), , Watch Video Solution, , 85., , Prove, , that:, 1, , (√3, , ×, , 5, , −3, , 3, , ÷ √3, , −1, , √5), , ×, , (3, , ×, , 6, , 5 ), , 6, , 3, =, 5, , Watch Video Solution

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86. Prove that: 9, , 3, , −, , 2, , 3, , ×, , 5, , 0, , −, , 1, −, , (, , 1, 2, , ), , 81, , = 15, , Watch Video Solution, , −2, , 1, , 87. Prove that: (, , 2, , ), , − 3, , ×, , 8, , ×, , 3, , 4, , 0, , −, , 9, +, , (, , 1, , 16, , 2, , ), , =, , 16, , 4, , 3, , Watch Video Solution, , 1, , 88. Prove that:, , 2, , 2, , 1, , ×, , 10, , −, , 3, , 3, , 1, , ×, , 4, , 1, 5, , 3, , ×, , 5, , 4, , 3, , 4, , ×, , 3, , ÷, 4, , 5, , −, , 5, , −, , 7, 5, , = 10, , 3, 5, , ×, , 6, , Watch Video Solution, , 89., , Prove, , that:, , (i) √, , 1, +, 4, , 2, , n, , (ii), 2, , +, , n+1, , 2, , n−1, , − 2, , n, , 3, =, 2, , Watch Video Solution, , (0. 01), , −, , 1, 2, , 3, , 2, , −, , (27) 3, , =, 2

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90. Prove that: (, , 2, , −, , 64, , √25, , 1, , 3, , ), , +, , +, , 1, , 125, (, , 256, 625, , (, , 65, ) =, 16, , 3, , √64, , 4, , ), , Watch Video Solution, , 3, , 91. Prove that:, 5, , 2, , ×, , −3, , 1, , 3, , √, , ×, , 6, ×, , 25, , 2, , ×, , √98, , (15), , −, , = 28√2, , 4, 3, , 1, , ×, , 3, , 3, , Watch Video Solution, , 0, , (0. 6), , 92. Prove that:, , −, , −1, , (, , 3, 8, , ), , (0. 1), , 1, 1 + x, , 3, 2, , ), , −1, , +, , ( −, , 1, a−b, , 3, , 3, , (, , Watch Video Solution, , 93., , −1, , +, 1 + x, , b −a, , Watch Video Solution, , = ?, , 1, 3, , ), , =, , −, 2

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94., , Show, , that, , x, , :, , x, (x, , a+b, , 2, , ) (x, , b +c, , )(x, , c +a, , a(b −c ), , x, ⋅, , b (a−c ), , x, , b, , a, , c, , ), , = 1, , 2, , ), , = 1, a, , b, , 4, , c, , (x x x ), , Watch Video Solution, , 95., , Show, , that:, , (i) (x, , a−b, , a+b, , ), , (x, , b −c, , b +c, , ), , (x, , c −a, , c +a, , ), , a, 1, , (ii) {(x, , a, , − a, , ( −1 ), , a +1, , a −1, , ), , }, , = x, , Watch Video Solution, , x +1, , 96. Show that: (, , a, , y +1, , x +y, , ), , y +2, , a, (, , a, , z+2, , y +z, , ), , z+x, , z+3, , a, (, , a, , x +3, , ), , a, , Watch Video Solution, , 97. Show that: (, , 3, , a, , 3, , b, , a+b, , ), , 3, (, 3, , b, , c, , b +c, , 3, ), , (, 3, , c, , a, , c +a, , ), , = 1, , = 1, , = 1

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Watch Video Solution, , 98. If 2x, , = 3, , y, , = 12, , show that, , z, , 1, , 1, , 2, , =, , +, y, , z, , x, , Watch Video Solution, , 99. If 2x, , = 3, , y, , = 6, , −z, , prove that, , 1, , 1, +, , x, , 1, +, , y, , = 0, z, , Watch Video Solution, , 100. If ax, , y, , = b, , z, , = c, , and b2, , = ac,, , then show that y, , 2zx, =, z + x, , Watch Video Solution, , 101. If 3x, , = 5, , y, , z, , = (75), , show that z, , xy, =, 2x + y, , Watch Video Solution

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102. If 27x, , 9, =, 3, , find x., , ,, , x, , Watch Video Solution, , 103. Find the values of x in each of the following : (i)25x, (ii)(23 ), , 4, , 2, , x, , = (2 ), , ÷ 2, , x, , = 25, , 20, , `, , Watch Video Solution, , 104., , Find, x, , 3, (i) (, , 5, , the, , ), , 2x, , 5, (, , 3, , value, , ), , of, , 125, =, , 27, , in, , x, , (ii) 5, , x −2, , each, ×, , 3, , of, , 2x − 3, , the, , following:, , = 135, , Watch Video Solution, , 105., 2, , x −7, , Find, ×, , 5, , the, x −4, , value, , = 1250, , of, , 2x +, , (ii) (4), , Watch Video Solution, , in, , x, , 1, , 1, 2, , each, , =, , 32, , of, , the, , following:

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106. Find the value of, √x, , (ii) (13), , = 4, , 4, , − 3, , 4, , x, , in each of the following:, , − 6 (iii) (√, , x +1, , 3, , ), 5, , 1, , = 2, , 3, , 2, , +, , 2, , show that x3, , 3, , − 6x = 6, , Watch Video Solution, , x, , 108. Determine (8x), , ,, , if 9x + 2, , = 240 + 9, , x, , Watch Video Solution, , 109. If 3x + 1, , = 9, , x −2, , ,, , find the value of 21 + x, , Watch Video Solution, , 125, =, 27, , Watch Video Solution, , 107. If x, , (i) 5, , 2x + 3, , = 1

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110. If 34x, , = (81), , −1, , and 10, , 1, y, , = 0. 0001,, , find the value of 2 − x + 4y, , Watch Video Solution, , 111. If 53x, , =, , 125, , and 10y, , = 0. 001, , find x, , and y, , Watch Video Solution, , 112., (ii)4, , Solve, 2x, , (iv) 8, , the, , = (163), , x +1, , = 16, , −, , following, 6, y, , equations:, , (i) 3, , 2, , = (√8), , (iii) 3, , 1, , and,, , x +2, , x −1, , (v) (, , x +1, , ×, , 4, , ×, , 3 − 2x, , (0. 5), , 1, = (, , x, , ), 8, , are distinct positive primes., Watch Video Solution, , a, , b, = (, , √, b, , ), , 2y − 3, , 3, , 4, , = 225, 1, , = (, , 3y, , ), 4, , 1 − 2x, , ), a, , 5, , ×, , 3 +x, , 2, x −1, , = 27, , , where, , a, b

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113., 6, , a b, , If, , −4, , are, , a and b, x, , 2y, , 3 = a b, , find x, , ,, , distinct, and y, , positive, , primes, , such, , that, , positive, , primes, , such, , that, , ., , Watch Video Solution, , 114., , If, a, , (i) (, , are, , a and b, , −1, , 2, , a b, , 7, , 2, , b, , ), , −4, , (ii) (a + b), , 3, , a, ÷ (, , a, , −1, , (a, , −1, , different, b, , −5, x, , −2, , + b, , 3, , find, , y, , ) = a b ,, , x and y, , b, , −1, , ) =, , x, , y, , a b ,, , find x, , + y + 2, , Watch Video Solution, , 115. If, , 2, , x, , value of 3x, , 3, , y, , 2, , 5, , z, , −y, , = 2160 ,, , 5, , find, , x, y and z, , . Hence, compute the, , −2, ., , Watch Video Solution, , 116. If, , 1176 = 2, , a, , ×, , 3, , b, , ×, , c, , 7 ,, , Hence, compute the value of 2a, , ×, , find the values of, 3, , b, , ×, , 7, , −c, , a, b and c, , as a fraction., , .

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Watch Video Solution, , 117., , x, , Simplify:, , (i) (, x, , 1, , x, (ii) (, x, , l, , m, , lm, , ), , x, ×, , x, , n, , x, , ), , c, , x, , mn, , ), , x, ×, , x, , a, , x, ), , c +a, , (, x, , b, , c −a, , ), , 1, , n, , (, , b −c, , b +c, , (, , 1, , m, , (, , a−b, , a+b, , ln, , ), , l, , Watch Video Solution, , (a +, , 118. Show that:, (b +, , 1, b, 1, a, , m, , ), , ×, , (a −, , m, , ), , ×, , (b −, , 1, b, 1, a, , n, , ), n, , ), , a, = (, , m +n, , ), b, , Watch Video Solution, , 119. If, m −n, , a, , a = x, n−1, , b, , m +n, , 1 −m, , c, , 1, , y , b − x, = 1, , Watch Video Solution, , n+l, , y, , m, , and, , c = x, , l+m, , y, , n, , ,, , prove that

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120., x, , m, , y, , If, n, , z, , l, , m +n, , x = a, n, , l, , = x y z, , n+l, , , y = a, , and, , m, , Watch Video Solution, , 121. Write (625), , −, , 1, 4, , in decimal form, , Watch Video Solution, , 122. State the product law of exponents., Watch Video Solution, , 123. State the quotient law of exponents, Watch Video Solution, , 124. State the power law of exponents, , l+m, , z = a, , ,, , prove, , that

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Watch Video Solution, , 125., x, , For, ), , x, , positive, , a+b, , a, , (, , any, , x, ×, , (, , b, , x, , number, , b +c, , b, , c, , real, , x, , ), , ×, , (, x, , find, , the, , value, , of, , the, , value, , of, , c +a, , c, , a, , x,, , ), , Watch Video Solution, , 126. Write the value of {5(8, , −, , 1, 2, , }, , −, , 2, , 1, 4, , }, , Watch Video Solution, , 127., , For, , any, , positive, , 1, , a, , b, , {(x ) }, , real, , 1, , ab, , b, , c, , bc, , {(x ) }, , c, , a, , {(x ) }, , number, , x;write, , 1, ca, , Watch Video Solution, , 128. If (x, , 3, , − 1), , = 8,, , what is the value of (x, , 2, , + 1), , ?

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Watch Video Solution, , 129. If 24, , ×, , 4, , 2, , then find the value of x ?, , x, , = 16 ,, , Watch Video Solution, , 130. If 3x − 1, , = 9, , and 4y + 2, , = 64, , , what is the value of, , x, ?, y, , Watch Video Solution, , 131. Write the value of √7, 3, , 3, , × √49, , Watch Video Solution, , 132. Write (, , −, , 1, ), , 1, 2, , ×, , (64), , 9, , Watch Video Solution, , −, , 1, 3, , as a rational number.

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133. Write the value of √125, 3, , ×, , 3, , √27, , Watch Video Solution, , 134. The value of {2, , 3, , − 3(2 − 3) }, , 3, , is 5 (b) 125 (c), , 1, , (d) − 125, , 5, , Watch Video Solution, , 0 . 16, , 135. (256), , 0 . 09, , × (256), , = ? 4, , b. 16 c. 64 d. 256. 25, , Watch Video Solution, , 136. If 102y, , =, , 25 ,, , then 10 − y equals −, , 1, 5, , Watch Video Solution, , (b), , 1, 50, , (c), , 1, 625, , (d), , 1, 5

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137. The value of, − 18, , x − y, , x −y, , when, , x = 2, , and, , y =, , − 2, , is 18 (b), , (c) 14 (d) − 14, Watch Video Solution, , 138. The product of the square root of, , x, , with the cube root of, , x, , cube root of the square root of x (b) sixth root of the fifth power of, , is (a), x, , (c), , fifth root of the sixth power of x (d) sixth root of x, Watch Video Solution, , 139. if 9x + 2, , x, , = 240 + 9 ,, , then x, , =, , 0. 5, , (b) 0.2 (c) 0.4 (d) 0.1, , Watch Video Solution, , 140. The seventh root of x divided by the eight root of x is x (b) √x (d), x56, , (d), , 1, x56, , Watch Video Solution

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141. The square root of 64 divided by the cube root of 64 is 64 (b) 2 (c), 1, , (d) 64, , 2, 3, , 2, , Watch Video Solution, , −, , 1, , 142. Which of the following is (are) not equal to, , 5, {(, , 1, 6, , 5, , ), , }, , ?, , 6, 1, 5, , 5, (a) (, , −, , ), , 1, , 1, , 6, , 1, (b), , 1, , 6, , 30, , 6, , 1, , (c) (, , ), 5, , 6, , −, , 5, (d) (, , 1, 30, , ), 6, , {(5) 5 }, , Watch Video Solution, , 143. When simplified, (c), , xy, x + y, , (d), , (x, , −1, , + y, , x + y, xy, , Watch Video Solution, , −1, , −1, , ), , is equal to, , (a) xy, , (b), , x + y

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144. If 8x + 1, , = 64,, , what is the value of 32x + 1 ? 1 (b) 3 (c) 9 (d) 27, , Watch Video Solution, , 145. If `0, , Watch Video Solution, , 146. If, 2√2, , x, , is a positive real number and, , x, , 2, , = 2,, , then, , (c) 3√2 (d) 4, Watch Video Solution, , 147. If (23 ), , 2, , x, , = 4 ,, , then 3x, , Watch Video Solution, , =, , 3 (b) 6 (c) 9 (d) 27, , x, , 3, , =, , √2, , (b)

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148. If, , 10, , x, , what is the value of, , = 64,, , x, , 10, , 2, , +1, , ? (a) 18 (b) 42 (c) 80, , (d) 81, Watch Video Solution, , x, , 149. If, x, , = 8x, , 1.5, , −1, , and, , x > 0,, , then, , √2, x =, , (b), , 2√2, , (c) 4 (d), , 4, , 64, Watch Video Solution, , 150. If g, , 2, , = t3 +, , 4t, , −, , 1, 2, , , what is the value of g when t, , = 64 ?, , Watch Video Solution, , 151. If x − 2, , = 64,, , then x, , 1, 3, , +, , x, , 0, , =, , (a) 2 (b) 3 (c), , 3, 2, , Watch Video Solution, , (d), , 2, 3

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152. If, , 4, , x, , −, , 4, , x −1, , = 24,, , then, , x, , (2x), , equals, , (b), , 5√5, , (c), , √5, , 25√5, , (d) 125, Watch Video Solution, , 153. When simplified (, , −, , 1, −, , 2, 3, , is 9 (b) − 9 (c), , ), 27, , 1, , 1, , (d) −, , 9, , 9, , Watch Video Solution, , 154. Which one of the following is not equal to, 8, , −, , 1, 6, , 1, , (c), , 1, , (83) 8, , (d), , (83), , −, , 1, 2, , ?, , (23), , −, , 1, , (b), , 2, , 1, √2, , Watch Video Solution, , 155. Which one of the following is not equal to, , −, , 100, (, , ), , 3, 2, , ?, , 9, 3, , 9, (a) (, , 1, , 2, , ), , (b), , 3, 3, , 100, (, , 100, 9, , 2, , ), , (c), , 3, ×, , 10, , 3, ×, , 10, , 10, , (d)

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√, , 100, , 100, x, , 9, , 100, x, , 9, , 9, , Watch Video Solution, , 156. When simplified (256), , − (4, , ( −3 / 2 ), , ), , is 8 (b), , 1, , (c) 2 (d), , 8, , 1, 2, , Watch Video Solution, , 157., , 5n + 2 − 6 × 5n + 113 × 5n − 25n + 1, , is equal to (a), , 53, , (b) − 53 (c) 35 (d) − 35, Watch Video Solution, , 158., , If, , √a − 1 b, , (d), , ×, , are, , a, b, c, √b − 1 c, , ×, , positive, , √c − 1 a, , 1, abc, , Watch Video Solution, , real, , is equal to:, , (a) 1, , numbers,, (b), , abc, , (c), , then, √abc

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159. If, , 3, , 2x − 8, , 5, =, , 225, , 5, , 3, , , then x, , x, , 2 (b) 3 (c) 5 (d) 4, , =, , Watch Video Solution, , 160. If (, , 2, , x, , ), 3, , 2x, , 3, (, , ), , 81, =, , 2, , . then x, , ,, , =, , 2 (b) 3 (c) 4 (d) 1, , 16, , Watch Video Solution, , 161. The value of {8, , −, , 4, 3, , ÷ 2, , −2, , 1, , )2, , is, , 1, , (b) 2 (c), , 2, , 1, , (d) 4, , 4, , Watch Video Solution, , 162. If, , a, b, c, , are positive real numbers, then, , equal to (a) 5a2 bc2 (b) 25ab2 c 5a3 bc3 (d) 125, Watch Video Solution, , 5, , √3125 a10 b5 c10, 2, , 2, , a bc, , is

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163. The value of 64 −, , 1, 3, , 1, , (64, , 3, , 2, , − 64, , 3, , ),, , is 1 (b), , 1, , (c) − 3 (d) − 2, , 3, , Watch Video Solution, , 164. If √5 n, , = 125,, , then 564n, , =, , 1, , 25 (b), , (c) 625 (d), , 125, , 1, 5, , Watch Video Solution, , 165. If, , 2x + 3, , (16), , x +3, , = (64), , ,, , then, , 4, , 2x − 2, , =, , 64 (b) 256 (c) 32 (d), , 512, Watch Video Solution, , 166. If a,, (b) a (c) a, , m, n, m, n, , are positive integers, then, , (d) 1, , Watch Video Solution, , {anm}, , mn, , is equal to, , mn, , a

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167. If, , 2, , −m, , 1, ×, 2, , equal to, , 1, , 1, , m, , =, , 1, , then, , ,, , {(4, , 4, , (b) 2 (c) 4 (d) −, , 2, , m, , 1/2, , ), , −1, , 1, +, , (, , 14, , 5, , m, , ), , }, , is, , 1, 4, , Watch Video Solution, , 168. If, , x = 2, , and, , y = 4,, , then, , x −y, , x, (, , y −x, , y, , ), , +, , (, , y, , ), , 4 (b) 8, , =, , x, , (c) 12 (d) 2, Watch Video Solution, , 1, 1, , ⎡, , 169. The value of, , m, , for which, , 1, , ⎢ {(, 7, , ⎣, , (b), , 1, , −2, , 2, , ), , 3, , }, , 4, , ⎤, ⎥, , = 7, , m, , ,, , is, , 1, −, 3, , ⎦, , (c) − 3 (d) 2, , 4, , Watch Video Solution, , 170. If, , 2, 2, , m +n, , m −n, , 2m + n − p, , 1, , = 16 and a = 2 10, , (a, , then, , m − 2n + 2p, , (a, , ), , 2, , − 1), , −1, , =

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Watch Video Solution, , 2, , 2, , 171. The value of, , 2, , {(23 + 2 ), , 3, , 1, , +, , (140 − 19), , 2, , }, , , is 196 (b) 289, , (c) 324 (d) 400, Watch Video Solution, , 172. If, , √2, , n, , = 1024,, , then 3, , 2(, , n, 4, , −4 ), , =, , 3 (b) 9 (c) 27 (d) 81, , Watch Video Solution, , 173. If, , 3, , 5x, , ×, , 81, 3, , (d) −, , 2, , ×, , 6561, , 2x, , 1, 3, , Watch Video Solution, , 7, , = 3 ,, , then, , x =, , (a) 3, , (b), , −3, , (c), , 1, 3

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174. Evaluate each of the following: (i)52, 3, , 11, (, , (v) (, , ), 12, , ⋅ 5, , 4, , (ii), , 5, , 8, , + 5, , 3, , (iii), , 2, , 3, , (3 ), , −3, , 3, ), 4, , Watch Video Solution, , 175. Evaluate each of the following: (i)(, , 4, , 2, , ), , ⋅ (, , 11, 1, (, , 5, , ), , 4, , −2, ⋅ (, , ⋅ (, , 3, , ), 3, , 3, ⋅ (, , 3, , ), , (ii), , 2, , −1, , 3, , ), , 2, , 2, , 11, , ), 5, , Watch Video Solution, , 176. If a, a, , a, , b, , + b, , then find the values of each of the following:, , = 2andb = 3,, , (ii) a, , b, , a, , + b, , (iii) a, , b, , (, , a, , a, , (v) (, , ), , b, , 1, , 1, +, , a, , a, , ), b, , Watch Video Solution, , 177. Assuming that, following: (i) √x, , −2, , x, y, , y, , 3, , are positive real numbers, simplify each of the, , (ii) (x, , −2, , . −, , y, , 1, 2, , 2, , ), , 5, , (iii)(√x − 3 )

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Watch Video Solution, , 3, , 178. Simplify :, , 3, , (25) 2 x(243) 5, 5, , 4, , (16) 4 x(8) 3, , Watch Video Solution, , 179. Simplify: (, , −, , 81, 16, , 3, , ), , −, , 25, , 4, , x[(, , 9, , ), , 3, , −3, , 5, , 2, , ÷ (, , ), , ], , 2, , Watch Video Solution, , 180., , If, , ., , √x, , −1, , are, , x, y, z, , y√y, , −1, , positive, , ., , z√z, , −1, , x = 1, , Watch Video Solution, , real, , numbers, , show, , that:

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181., , Show, , that, , x, , :, , x, (x, , a+b, , 2, , ) (x, , b +c, , )(x, , c +a, , a(b −c ), , x, ⋅, , b (a−c ), , x, , c, , b, , a, , ), , = 1, , 2, , ), , = 1, a, , b, , c, , 4, , (x x x ), , Watch Video Solution, , n, , 2, , 9 x3 x(3, , 182., , −, , If, 3, , −2, , n, 2, , 3m, , ), , n, , − (27), , x2, , 1, =, , 3, , Prove, , ,, 27, , m − n = 1., , Watch Video Solution, , a, , 183. Prove that :, a, , −1, , −1, , + b, , a, −1, , +, a, , −1, , −1, , − b, , 2, , 2b, −1, , =, , Watch Video Solution, , 184. Find the value of x, if 5x − 3 . 32x − 8, Watch Video Solution, , = 225, , 2, , b, , 2, , − a, , that

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185. If 27x, , 9, =, 3, , x, , find x., , ,, , Watch Video Solution, , 186. Find the values of x in each of the following : (i)25x, (ii)(23 ), , 4, , 2, , = (2 ), , x, , `, , Watch Video Solution, , ÷ 2, , x, , = 25, , 20