Quantitative Aptitude-More exercise questions updated on Jan 2022
Surds are numbers left in 'square root form' or 'cube root form'. A surd is the root of a whole number that has an irrational value. Some examples are ?2 ?3 ?10. All surds are irrationals but all irrational numbers are not surds. An index number is a number which is raised to a power. The power, also known as the index, tells you how many times you have to multiply the number by itself.

Surds and Indices-Exercise Questions

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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  x9/2 . √y7 is:

x7/2 . √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 10x exceeds 1000.

a. 8

b. 1

c. 7

d. 6

5. Which among the following is the greatest?

a. 23^2

b. 22^3

c.  32^3

d. 33^3

6. Solve for m if 49(7m) = 3433m + 6

a. -8/6

b. -2

c. -4/6

d. -1

7. Solve for 2y^√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, 2a = b3 and ba = 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  44m + 2 = 86m – 4, solve for m.

a. 7/4

b. 2

c. 4

d. 1

11. If  2x  x 162/5 = 21/5, then x is equal to:

a. 2/5

b. -2/5

c. 7/5

d. -7/5

12. If ax = by = cz 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 7a = 16807, then the value of 7(a – 3) is:

a. 49

b. 343

c. 2401

d. 10807

14. If 3x – 3x – 1 = 18, then the value of xx 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 ax = b, by = c and cz = 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 52n – 1 = 1/(125n – 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

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

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

2. Exp: x9/2 . √y5 is: = x(9/2 – 5/2) . y(7/2 – 3/2) = x2. y2

x7/2 . √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 10x exceeds 1000 when x has a minimum value of
8.

5. Exp: 23^2  = 29

22^3 = 28

32^3 = 38

33^3 = 327

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

6. Exp: 49(7m) = 3433m + 6 Þ 727m Þ (73)3m + 6  Þ 72 + m = 79m + 18

Equating powers of 7 on both sides,

m + 2 = 9m + 18

-16 = 8m Þ m = -2.

7. Exp: 3y^√2^2 = 729

3y^2 = 34 (√22 = (21/2)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: 2a = b3 ….(1)

ba = 8 …..(2)

cubing both sides of equation (2),(ba)3 = 83

b3a = (b3)a = 512.

from (1),(2a)a = (23)3.

comparing both sides, a = 3

substituting a in (1),b =2.

10. Exp: 44m + 2 = (23)6m – 4 => 44m + 2 = 218m – 12

Equating powers of 2 both sides,

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

11. Exp:  2x x 162/5 = 21/5

=> 2x x (24)2/5 = 21/5 => 2x x 28/5 = 21/5.

=> 2(x + 8/5) = 21/5

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

12. Exp: Let ax = by = cz = k. Then, a = k1/x, b = k1/y, c = k1/z.

Therefore,  b² = ac => (k1/y)2 = k1/x x k1/z  =>  k2/y = k(1/x + 1/z)

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

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

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

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

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

2(x + y) = 32 = 2=> x + y = 5 ---(2)

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

16. Exp: a1 = cz = (by)z = byz = (ax)yz = axyz.   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: 52n – 1 = 1/(125n – 3)  => 52n – 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.

Surds are numbers left in 'square root form' or 'cube root form'. A surd is the root of a whole number that has an irrational value. Some examples are ?2 ?3 ?10. All surds are irrationals but all irrational numbers are not surds. An index number is a number which is raised to a power. The power, also known as the index, tells you how many times you have to multiply the number by itself. Freshersworld.com explains basic concept of Surds and indices and provides aptitude test, shortcuts, tricks, and formulas. It also provides questions and answers, practice questions and solved examples that would help candidates in clearing all the competitive tests. It also provides online test on Surds and indices - quantitative Aptitude. What are the rules of a surd? Some of the important rules are mentioned below  Every rational number is not a surd.  Every irrational number is a surd.  A root of a positive real quantity is called a surd if its value cannot be exactly determined.  ?a × ?a = a ? ?5 × ?5 = 5  If a and b are both rationals and ?x and ?y are both surds and a + ?x = b + ?y then a = b and x = y  If a  ?x = b  ?y then a = b and x = y.  If a + ?x = 0, then a = 0 and x = 0.  If a  ?x = 0, then a = 0 and x = 0. What are the rules of indices? a0=1 a-m=1/am am*an=am+n am/an=am-n (am)n=amn a1/a1=a0
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