1. If m and n are whole numbers such that mⁿ = 169, then the value of (m – 1) ^{n + 1} is:

a. 1

b. 13

c. 169

d. 1728

2. The simplified form of  x^9/2 . √y⁷ is:

                                     x^7/2 . √y³

a. x²/y²

b. x² . y²

c. xy

d. x²/y

3. If  √(3 + ³√x) = 2, then x is equal to :

a. 1

b. 2

c. 4

d. 8

4. If x is an integer, find the minimum value of x such that 0.00001154111 x 10^x exceeds 1000.

a. 8

b. 1

c. 7

d. 6        

5. Which among the following is the greatest?

a. 2^{3^2}

b. 2^{2^3}

c.  3^{2^3}

d. 3^{3^3}

6. Solve for m if 49(7^m) = 343^{3m + 6}

a. -8/6

b. -2

c. -4/6

d. -1

7. Solve for 2^{y^√2^2} = 729.

a. ±3

b. ±1

c. ±2

d. ±4

8. √[200√[200√[200……..∞]]] = ?

a. 200

b. 10

c. 1

d. 20

9.  If a and b are positive numbers, 2^a = b³ and b^a = 8, find the value of a and b.

a. a = 2, b = 3

b. a = 3, b = 2

c. a = b = 3

d. a = b = 2

10. If  4^{4m + 2} = 8^{6m – 4}, solve for m.

 a. 7/4

b. 2

c. 4

d. 1

11. If  2^x  x 16^2/5 = 2^1/5, then x is equal to:

a. 2/5

b. -2/5

c. 7/5

d. -7/5

12. If a^x = b^y = c^z and b² = ac, then y equals :

a. xz/x + z

b. xz/2(x + z)

c. xz/2(x – z)

d. 2xz/(x + z)

13. If 7^a = 16807, then the value of 7^{(a – 3)} is:

a. 49

b. 343

c. 2401

d. 10807

14. If 3^x – 3^{x – 1} = 18, then the value of x^x is:

a. 3

b. 8

c. 27

d. 216

15. If 2^{(x – y)} = 8 and 2^{(x + y)} = 32, then x is equal to:

a. 0

b. 2

c. 4

d. 6

16. If a^x = b, b^y = c and c^z = a, then the value of xyz is:

a. 0

b. 1

c. 1/abc

d. abc

17. 125 x 125 x 125 x 125 x 125 = 5^?

a. 5

b. 3

c. 15

d. 2

18. If 5^{2n – 1} = 1/(125^{n – 3}),then the value of n is:

a. 3

b. 2

c. 0

d. -2

19. If x = 5 + 2√6, then (x – 1) is equal to:

                                          √x

a. √2

b. 2√2

c. √3

d. 2√3                    

20. Number of prime factors in 6¹² x (35)²⁸ x (15)¹⁶ is :

                                                       (14)¹² x (21)¹¹

a. 56

b. 66

c. 112

d. None of these

Answer & Explanations

1. Exp: Clearly, m = 13 and n = 2.

           Therefore,  (m – 1) ^{n + 1} = (13 – 1)³ = 12³ = 1728.

2. Exp: x^9/2 . √y⁵ is: = x^{(9/2 – 5/2)} . y^{(7/2 – 3/2)} = x². y²

           x^7/2 . √y³

3. Exp: On squaring both sides, we get:

           3 + ³√x = 4 or ³√x = 1.

          Cubing both sides, we get x = (1 x 1 x 1) = 1

4. Exp: Considering from the left if the decimal point is shifted by 8 places to the right, the number
          becomes 1154.111. Therefore, 0.00001154111 x 10^x exceeds 1000 when x has a minimum value of
          8.

5. Exp: 2^{3^2}  = 2⁹

           2^{2^3} = 2⁸

           3^{2^3} = 3⁸

           3^{3^3} = 3²⁷

           As 3²⁷ > 3⁸, 2⁹ > 2⁸ and 3²⁷ > 2⁹. Hence 3²⁷ is the greatest among the four.

6. Exp: 49(7^m) = 343^{3m + 6} Þ 7²7^m Þ (7³)^{3m + 6}  Þ 7^{2 + m} = 7^{9m + 18}

           Equating powers of 7 on both sides,

           m + 2 = 9m + 18

           -16 = 8m Þ m = -2.

7. Exp: 3^{y^√2^2} = 729

          3^{y^2} = 3⁴ (√2² = (2^1/2)² = 2)

          equating powers of 2 on both sides,

          y² = 4 Þ y = ±2

8. Exp: Let √[200√[200√[200……..∞]]] = x ; Hence √200x = x

            Squaring both sides 200x = x² Þ x (x – 200) = 0

           Þ x = 0 or x – 200 = 0 i.e. x = 200

           As x cannot be 0, x = 200.

9. Exp: 2^a = b³ ….(1)

           b^a = 8 …..(2)

           cubing both sides of equation (2),(b^a)³ = 8³

           b^3a = (b³)^a = 512.

           from (1),(2^a)^a = (2³)³.

           comparing both sides, a = 3

           substituting a in (1),b =2.

10. Exp: 4^{4m + 2} = (2³)^{6m – 4} => 4^{4m + 2} = 2^{18m – 12}

            Equating powers of 2 both sides,

            4m + 2 = 18m – 12 => 14 = 14m => m = 1.

11. Exp:  2^x x 16^2/5 = 2^1/5

             => 2^x x (2⁴)^2/5 = 2^{1/5 =>} 2^x x 2^8/5 = 2^1/5.

             => 2^{(x + 8/5)} = 2^1/5

             => x + 8/5 = 1/5 => x = (1/5 – 8/5) = -7/5.

12. Exp: Let a^x = b^y = c^z = k. Then, a = k^1/x, b = k^1/y, c = k^1/z.

             Therefore,  b² = ac => (k^1/y)² = k^1/x x k^1/z  =>  k^2/y = k^{(1/x + 1/z)}

             Therefore,  2/y = (x + z)/xz => y/2 = xz/(x + z) => y = 2xz/(x + z).

13. Exp: 7^a = 16807, => 7^a = 7⁵, a = 5.

            Therefore,   7^{(a – 3)} = 7^{(5 – 3)} = 7² = 49.

14. Exp: 3^x – 3^{x – 1} = 18 => 3^{x – 1} (3 – 1) = 18 => 3^{x – 1} = 9 = 3² => x – 1 = 2 => x = 3.

15. Exp: 2^{(x – y)} = 8 = 2³  => x – y = 3 ---(1)

             2^{(x + y)} = 32 = 2⁵  => x + y = 5 ---(2)

             On solving (1) & (2),we get x= 4.

16. Exp: a¹ = c^z = (b^y)^z = b^yz = (a^x)^yz = a^xyz.   Therefore,    xyz = 1.

17. Exp: 125 x 125 x 125 x 125 x 125 = (5³ x 5³ x 5³ x 5³ x 5³) = 5^{(3 + 3 + 3 + 3 + 3)} = 5¹⁵.

18. Exp: 5^{2n – 1} = 1/(125^{n – 3})  => 5^{2n – 1} = 1/[(5³)^{n – 3}] = 1/[5^{(3n – 9)}] = 5^{(9 – 3n)}.

                                         => 2n – 1 = 9 – 3n => 5n = 10 => n = 2.

19. Exp: x = 5 + 2√6 = 3 + 2 + 2√6 = (√3)² + (√2)² + 2 x √3 x √2 = (√3 + √2)²

            Also, (x – 1) = 4 + 2√6 = 2(2 + √6) = 2√2 (√2 + √3).

           Therefore,  (x – 1) = 2√2 (√3 + √2) = 2√2.

                              √x           (√3 + √2)

20. Exp: 6¹² x (35)²⁸ x (15)¹⁶  = (2 x 3)¹² x (5 x 7)²⁸ x (3 x 5)¹⁶ =

               (14)¹² x (21)¹¹                 (2 x 7)¹² x (3 x 7)¹¹

            = 2¹² x 3¹² x 5²⁸ x 7²⁸ x 3¹⁶ x 5¹⁶  = 2^{(12 – 12)} x 3^{(12 + 16 – 11)} x 5^{(28 + 16)} x 7^{(28 – 12 – 11)}

                         2¹² x 7¹² x 3¹¹ x 7¹¹

            = 2⁰ x 3¹⁷ x 5⁴⁴ x 7^-5 = 3¹⁷ x 5⁴⁴

                                                  7⁵

             Number of prime factors = 17 + 44 + 5 = 66.