SURDS (COMPETITION)

SURDS

Any number of the form p/q, where p and q are integers and q≠ 0 is called a rational number. Any real number which is not a rational numbers, of particular interest to us are SURDS  Amongst surds, we will specifically be looking at 'quadratic surds' :- shrds of the type a+√b and a+ √b + √c, where the terms involve only square roots and any higher roots. We do not need to go very deep into the area of surds - what is required is a basic understanding of some of the operation on surds.

If there is a surd of the form (a+√b), then a surd of the form ±(a - √b) is called conjugate of the surd (a+√b). The product of a surd and its conjugate will always be a rational number.

 Rationalisation of a surd

 When there is a surd of the 1/(a+√b), it is difficult to perform arithmetical operations on it. Hence, the denominator is converted into rational number thereby facilitating ease of handling the surd. This process of converting the denominator into a rational number without charging the value of the surd is called rationalisation.

        To convert the denominator of a surd into a rational number, multiply the denominator and the numerator simultaneously with the conjugate of the surds in the denominator so that the denominator gets converted to a rational number without changing the value of the fraction. That is, if there is a surd of the type a+√b in the denominator, then both the numerator and the denominator have to be multiplied with a surd of the form a- √b or a surd of the type (-a-√b) to convert the denominator into a rational number.

 If there is a surd of the form (a+√b +√c) in the denominator, then the process of multiplying the denominator with its conjugate surd has to be carried out TWICE to rationalise denominator.

SQUARE ROOT OF A SURD

If they are exists a square root of a surd of the type a+√b, then it will be of the form √x +√y. We can equate the square of  √x +√y to √a +√b and thus solve for x and y. Here, one point should be noted :- when there is an equation with rational and irrational terms, the rational part of the left hand side is equal to the irrational part on the right hand side and, the irrational part on the right hand side of the equation.

 However, for the problems which are expected in the competitive exams, there is no need for solving for the square root in such an elaborate manner. We will look at finding the square root of the surd in a much simpler manner. Here, first the given surd is written in the form of (√x +√y)² or (√x -√y)². Then the square root of the surd will be (√x +√y) or (√x -√y) respectively.

 Comparison of surds

Sometimes we will need to compare two or more surds to identify the largest one or to arrange the given surds in ascending/descending order. The surds given in such cases will be such that they will be close to each other and hence we will not be able to identify the largest one by taking the approximate square root of each of the terms. In such a case, the surds can be both be squared and the common rational part be subtracted. At thid stage, normally one will be able to make out the order of surds. If even at this stage, it is not possible to identify the larger of the two, then the number should be squared once more.

Some Identities Related to Real Numbers:

1) (√a)²= a

2) √(ab)= √a √b

3) √(a/b)= √a/√b

4) (√a+√b)(√a - √b)= a - b

5) (a+√b)(a - √b)= a² - b

6) (√a+√b)(√c + √d)= √(ac)+ √(ad) + √(bc) + √(bd)

7) (√a- √b)² = a - 2√(ab) + b

1) Express 5 as a biquadratic surd.

A) √5 B) ³√5 C) ⁴√5 D) 5

Express in simplest rationalisation factor of:

2) √32

A) 4 B) √2 C) 4 √2 D) 2√2 

3) ³√49 

A) 7. B) √7 C) 14 D) none

4) ⁴√25

A) 5 B) √5 C) 2√5 D) n

5) ⁵√(a²b³c⁴)

A) abc B) a²bc C) a²b²c² D) ab³c³

6) ⁵√486

A) ⁵√16 B)  3 C) √3 D) 6

7)  ³√1080 

8) ³√192

9) ⁴√1280

Arrange in ascending as well as descending order of magnitude 

10) √5 , ²√11, 2 ⁶√2

11) ⁴√3, ⁵√4, ¹⁰√12

12) 2√2, 2 ³√2 , 2 ⁶√5

Add:

13) (2√3 +5√5 - 7√7) and (3√5 - √3 + √7)

A) 3√3+ √5- 6√7 

B) √3+ 8√5- 6√7.

C)  3√3+ √5- √7 

D) √3+ √5- 6√7 

14) (4√3 + 7√2) and (√3 - 5√2)

A) 3√3+ √2        B) 3√5+ 8√5

C) 5√3+ 2√2.     D) √3+ √2

15) (√5 +2√3) and (2√5 - 5√3)

A) 3√5 - 3√3.     B) √5+ 6√5

C)  3√3+ √5       D) 2√3+ √5

16) (-√2/2 +2√5/3 + 6√7) and (√5/3 + 3√3/2 - √7)

A) √2+ √5 +5√7. 

B) √3+ 8√5- 6√7

C)  3√3+ √5- √7 

D) √3+ √5- 6√7 

Multiply:

17) 7√6 by 5√24

A) 20 B) 240 C) 420. D) 460

18) ³√32 by ³√250

A) 20. B) 240 C) 420 D) 460

19) ³√7 by √2

A) ⁶√392 B) ⁵√392 C) ⁴√392 D) n

20) 2√3 by  5√27

A) 9 B) 90. C) 900 D) 9000

21) 3√28 by 2√7

A) 8 B) 4 C) 48 D) 84.

22) 3√8 by 3√2

A) 63 B) 36. C) 3 D) 6

23) 4√12 by  7 √6

A) 16 √2  B) 18√2 C) 168√2. D) 6√2

24) ³√2 by ⁴√3

A) ¹²√42 B) 6√42 C) 4√42 D) ¹²√432

25) 2 ⁴√3 by 5 ⁴√81

A) 30 ⁴√3.          B) 15 ⁴√3

C) 5 ⁴√3             D) 8 ³√3

Divide:

26) ³√18 by  ³√9

A) ³√2 B) ³√3 C) ³√9 D) ³√4

27) ³√128 by ⁵√64

A) 2 ¹⁵√4. B) ¹⁵√4 C) ¹⁵√2 D) none

28) 12√15 by 4√3

A)  √5 B) 3√5. C) 2√3 D)  3√7

29) 4√28 by 3√7 

A) 2/3 B) 4/3 C) 8/3. D) 10/3

30) 21√384 by 8√96

A) 21/4. B) 22/3 C) 41/3 D) 1

31) ⁶√12 by 3 ³√2

A) ³√(1/3). B) √4 C) √3 D) 6

Simplify:

32) (√147 - √108 - √3) 

A) 1 B) 0 C) 2 D) none

33) 4√3 - 3√12 + 2√75

A) 8√3. B) √3 C) 2√3  D) 3√3

34) 3√45- √125 + √200 - √50

A) 4√5 B) 5√2 C) 4 √5 + 5√2 D) 4√5 - 5√2

35) 3√48 - 5/2 √(1/3)+ 4 √3 

A) 91√3/6. B) ) 3√3 C) √5 D) 6√7 

36) 2 ³√4 + 7 ³√32 - ³√500

A) 11 √4 B) 11³√4. C) 11⁴√4 D) n

37) 2 ³√54 + 3 ³√16 + 5 ⁴√128

A) 32 ³√2.        B) 30 ³√2

C) 32 ³√3        D) 3√2

38) √125 - 4√6 + √294 - 2 √(1/6)

A) 5√5 B) √3+ 8√5 C) 6√7 D) none

39) ⁴√81 - 8 ³√216 + 15 ⁵√32 + √225

A) 0 A) 1 C) 2 D) 3

40) (5 +√3)(7 +√5)

A) 35 + 7√3 +5√5 +√15.

B) 3√5 - √3 + √15

C) 3√3+ √5- 6√5 

D) √3+ 8√5- 6√5

41) (3 + √2)(4+ √3)

A) 12 + √2+ √3+ √6

B) 12+ 4√2 +3√3 + √6

C)  12+ 3√2 - 3√3 + √6

D) ) 3√3+ √5- 6√6

42)  (√5+ √2)(√3+ √2)

A) √15- √10 +√6+ 2.

B) √5- 6√10 +√6

C) 2√5 +5√5 - 7√6

D) 3√5 - √5 + √2

43)(√13+ √11)(√13- √11)

A) 1 B) 0 C) 2 D) none

44) (4+ √3)(4 - √3)

A) 0 B) 1 C) 3 D) 13

45) (√13 - √6)(√13+ √6)

A) 3 B) 4 C) 7.

46) (3√5 +3√7)(3√5 - 2√7) 

A) 12  B) 12 + 2√35. C) 1 D) 5

47) (3√5 + 5√2)²

A) 95 + √10    B) 95+ 30√10.

C) 95 - √19     D) 95 +√3

48) (√5 + √7)²

A)  12 + 2√35. B) 12+ 2 √5

C) 12- 2√7       D) 12 + 2√3

49) (4√3 - 3√5)²

A) 93 + 24√15     B) 90 - √15

C) 93 - 24√15      D) none

50) (2√5/3 + √2/2 + 6√12) + (√5/3 + 3√2/2 - √11)

A) √5+ √2+ 5 √11

B) √5/2 + 2√2+ √11

C)  √5+ √2+ 6√11

D) √5+ 2√2+ 5 √11

51) ³√7 x √5

A) ³√35 B) ⁶√35 C) ⁶√6125 D) ⁶√1225

52)  √18/6 x+ √18/3

A) 1 B) 1/12 C) 1/3 D) √2

53) √5x √7 x √15 x √21

A) √105 B) √210 C) 105 D) 210

54) (3+ √3)(3 - √3)

A) 18 B) 2√3 C) 6 D) 9

55) (3+ √5)²(3- √5)²

A) 15 B) 16 C) 4 D) 14

56) ³√250 ÷ ³√10

A) ³√25  B) 5 C) √5 D) ³√2500

57) 30/(√20+√5)

A) 10/3√5  B) 30/√5 C) 10/√5 D) 12√5

58) 6/(√12+√3)

A) 1/√3  B) 2/√3  C) 2√3 D) 6√3

Simplify

59) ³√24 - ³√192 + ³√81

60) (√5 +√3)²(4 - √15)

61) 3√48 - 4√75 + 5√192

62) ³√56 - ³√875 + ³√189

63) √(x³y) + √(xy³) + √(xyz²)

64) (1+√2 - √3)(2 + √2 + √6)

65)(√50+√32-√18)/(√75 -3√3 +√12)

66) Express 4 as a cubic surd

67) Express x²y as fifth order surd

68) Find the square of √(x+y)-√(x-y)

69) Find the cube of 2√3 - 3√2

B) Rationalise the denominator

70) 3/√2                                   3√2/2

71) 3/(2+√3).                        3(2- √3)

72) 2/(√5 - √3)                     (√5+√3)

73) 3/(4√5 - 2).           3(2√5 +1)/38

74) 5/(4√5 - 5√3).           (4√5+5√3)

75) (3√2+2√3)/(3√2 -2√3).  (5+2√6)

76)√3/(√2+√3 -√5).    (2+√6+√10)/4

77) 1/(1 + √2 - √3).       (√2+2+√6)/4

78) 12/(3 +√5 + 2√2).         (1+√5+√2-√10)

79) (√3 - √2)/(√3 + √2).            5-2√6

C) Simplify 

80) 1/(√6 -√5) - 3/(√7 - √2) - 4/(√6 + √2).                                 2√2+ √5+ √7

81) √5/(√3 +√2) - 3√3/(√2 + √5) + 2√2/(√5 +√3).                           0

82) If x= √{(√5 +1)/(√5 -1)}, Show that x² - x -1=0

83) If x= √2 +√3, show that x⁻² = 5 - 2√6

84) If a= √3/ 2, show that √(1+a) +√(1- a) = 2a

85) If x = 4+ 2√3, find the value of (√x - 2/√x)

86) If x = 4 + √15, Show that √x + 1/√x = √10

E) Find the value of

87) √[2 +√{2 + √(2 + ....... up to ∞)}]

88) √[³√{b √(a ³√b ..... up to ∞)}]

89) √[6 + √{6 +√(6+ .....up to ∞)}]

EXERCISE -1

1) Simplify: 1/(4-√5) - 1/(4+ √5)

A) 2 B) √5 C) 2√5 D) 2√5/11 E) n

2) Rationalise the denominator: 1/(1+ (√6 - √7).

A) (√6+ 6+ √42)/12

B) (6+ √6+ √42)/12

C) (√6- 6+ √42)/12

D) (√6- 6- √42)/12.  E) n

3) Find the value of √{62+ √480}

A) √60+√2 B) √60- √2 C) 60+√2 D) √60+ 2 E) n

4) Which of the surds is greater? √3+ √23 and √6+ √19.

A) √6+ √19 > √3+ √23 

B) √3+ √23 > √6+ √19

C) both are equal

D) not equable

5) Which of the following is the conjugate of the surd √7- 2 ?

A) √7+ 2 B) -2 -√7 C) either A or B D) n

6) simplify: 3/(√5+ √2) + 1)(√6+ √5)

A) √6-√2 B) √6+ √2 C) √5+ √6 D) √2- √6 E) n

7) Find the positive square root of √180 - √125

A)√5 B) 5¹⁾⁴ C) 65¹⁾⁴ D)180¹⁾⁴ -125¹⁾⁴ E) n

8) Find the positive square root of 27- 10√2.

A) 5- √2 B) √200- √27 C) 6- √2 D)  4- √5  E) n

9) 9/(6²⁾³ - 18¹⁾³ + 3²⁾³) =

A) (6¹⁾³ + 3¹⁾³) B) (1/3)(6¹⁾³+3¹⁾³) C) (6¹⁾³ - 3¹⁾³) D) (1/9)(6¹⁾³+ 3¹⁾³) E) n

10) Which of the following is a rationalising factor of 10¹⁾³ - 9¹⁾³ ?

A) 10²⁾³ + 9²⁾³) B) 10²⁾³ - 9²⁾³) C) 10²⁾³ + 90¹⁾³ + 9² D) 19²⁾³ - 9¹⁾³ + 9²⁾³ E) n

11) Which of the following is the rationalising factor of 12 + 12√6 ?

A) -12 - 12√6 B) -12 - 12√6 C) 12 - 12√6 D) either B or C E) n

12) Find the square root of √98 + √96

A) ⁴√2(2+ √3) B) √2(√2 + √3) C) √2(2+ √3) D) ⁴√2(√2 + √3) E) n

13) Arrange the following in ascending order a= √2 + √11, b= √6 + √7, c= √3 +  √10 and d= √5+ √8

A) abcd B) adbc C) acdb D) acbd E) n

14) Arrange the following in descending order a= √20 + √2, b= √24 + √6, c= √22 + 2 and d= √26+ √8

A) dcba B) dcab C) dbca D) dbca E) n

15) a= √3 + √23, b= √6 + √19, is greater than b ?

A) true B) false C) cannot say 

16) Arrange the following in descending order a= √13 + √11, b= √15 + √9, c= √18 +  √6 and d= √7+ √17

A) abdc B) dcab C) adcb D) acdb E) n

17) Arrange the following in ascending order p= √26 - √23, q= √18 - √15, r= √11 -  √8 and s= √24 - √21

A) rqsp B) psrq C) pqrs D) psqr E) n

18) Simplify: = [a/(√b - √c) + a/(√b + √c)]²

A) 2a²c/(b- c)² B) (b- c)²/4a²b  C) 4a²b/(b- c)²  D) 2a²/(b²+c² - a²)  E) 2ac/(b- c)² F) n

19) If 2√2 + √3= x, what is the value of (11+ 4√6)/(2√2 - √3) in terms of x ?

A) x²/√2 B) x³ C) x³/8 D) x E) x³/5

20) If √[x {√x (√x.......∞)= 11ˣ, what is the value of (ˣ√x) ?

A) ³√11 B) 1 C) √11 D) 11¹⁾ˣ E) 11

21) If 1/(√x + √(x+1)) + 1/(√(x+1) + √(x+2)) + 1/(√(x +2)+ √(x+3))+ 1/(√(x +98)+ √(x+99)) = 9, find which of the following is a positive value of x.

A) 2 B) 1 C) 4 D) 3 E) 5

22) What is the cube root of (3√3)(2√2)+ 7√7 + 3√3)(√2)(√7)(√3 √2+ √7) ?

A)√3 + 7√2 B) 6+ √7 C) √6 + 7√7 D) √6 + √7 E)  7 + √6

23) Find the value of  √(16 + 2√55).

A) 1+ √15 B) √10 + √5 +5 C) 1+ 2√5 + √15 D) √12 + √5 E) √11+ 3√5

24) Simplify: √{(a+b+c) + 2√(ac+ bc)}.

A) √a + √b +√c B) √(a+b) + √c D) √(ab+ bc) D) √(abc) E) √(ac+ bc)

25) Arrange a, b, c, d in ascending order if a= √13 + √9, b= √19 + √3, c= √17 +  √5 and d= √12+ √10

A) b,c, a,d B) d,b,a,c C) d,b,c,a D) d, c, b, a E) d,a,c,b

26) if a= 4(b-1) and b≠ 2, then simplify: √{b - √a)/(b+ √a)} + {b+ √a)/(b -  √a)} - 4/{(b- √a)(b +  √a)} 

A) √b B) 2 C) b/(b-2) D) (b-2)¹⁾² E) 4

27) Simplify: (a- b)/{³√a² + ³√(ab) + ³√b²} - (a+ b)/{³√a² - ³√(ab) + ³√b²}.

A) -2b¹⁾³ B) -2(ab)¹⁾³ C) a¹⁾³ - b¹⁾³ D) (a/b)¹⁾³ E) (b/a)¹⁾³

28) If √[x+ √{x²+ √(x⁴+√(x⁸.......∞).

A) √x{(1+√5)/2}  B) {(3+ 5√2)/√x}  C) x)(1+√x) D) (√2+√3)/x³⁾²   E) (√3+√5)/x³⁾²

29) If A/a = B/b = C/c = D/d, what is the value of √(Aa) + √(BB) + √(Cc) + √(Dd), given that (a+ b+c+ d)≠ 0 ?

A) 1 B) (A+B+C+D)/(a+b+c+ d)

C) =(√A+ √B+ √C+ √D)(a+b+c+d)

D) √{(a+b+c+d)(A+B+C+D)

E) √{(A+B+C+D)/(a+b+c+d)}

30) √(2x²+9) + √(2x²- 9)= 9+ 3√7 and √(2x²+9) - √(2x²- 9)= √a- √b, find the values of a and b

A) 18, 36 B) 81, 63 C) 27, 96 D) 34, 28 E) 63, 18

31) Find the square root of 23+ 4√10 - 10√2- 8√5.

A) √5 + √10 - √8 B) √10 + √8 - √5 C) √8 + √5 + √10 D) √5 + √8 - √10 E) √10 - √5 - √8 

32) a, b, c and d are rational numbers such that 16/2+ √5 + √13)= a+ b √5 + c √13 + d√65. Find the value of abcd 

A)21 B) 25 C) 28 D) 35 E) 42

33) What is the mean proportional of 2 - √3 and 26 - 15 √3 ?

A) 7  - 4√3 B) 12- 7√3 C)!13- 5√3 D) 24√3 - 14√3 E) 12 - 4 √3

34) Find the square root of [1+ 1/(√2 +1) + 1/(√3 + √2) + 1/(√4+ √3)  + 1/(√324 + √323)]

A) 3√2 B) 1/√2 C) 2 √3 D) (√5 -2)/2 E) 1/√3

35) If a= √6 + √8, what is 2 √2(26+ 15√3) in terms of a ?

A) √a³ B) √a⁵  C) a⁴ D) a³ E) a²

36) solve: √x + √{x - √(1- x)}= 1.

A) 1 B) 16/25 C) 4/5 D) 0 E) 5/4

37) value of √(7 - 3√5)

A) √10 - 2√3 B) (3-√5)/√2 C) (√3 - √7)/2√2 D) (√5 + √2)/3 E) 5 - √5)/2

38) Which of the following is a rationalising factor of (⁸√3 - ⁸√2)(⁴√3 + ⁴√2)(√3 - √2) ?

A) (⁴√3 - ⁴√2( B) (⁸√3 + ⁸√2) C) (1- √3 - √2) D) (√3 - √2)(⁴√3  - ⁴√2) E) (√3 - √2)

39) if a= 1/(2+√3), b= 1/(2- √3), what is the value of 7b² + 11ab  - 7a²

A) -14 + 21√3 B) 11 + 56√3 C) 49 + 8 √3 D) √3 + √11 E) 11 - √3

40) The arithmetic mean of two surds is 5+ 9√2, and one of the surds is 1+ 12√2. What is the square root of the other surd ?

A) 6 - 21√2 B) 4- 3√2 C) √3(√2+ 1) D) √2(2 - √3) E) 2(√3 - √2)

41) If x= 3 - √5 then the value of √1x/{√2 + √(3x- 2)}

A) 1/(4 - 2√5) B) √3 - 2√5 C) 1/√5 D) 1/(√2- 3 √5) E) 1/(√2 + 3√5)

42) if x= 5 - √21, find the value of √x/{√32 - 2x) - √21}

A) (√7 - √3)/2 B) (√7 - √3)/√3 C) (√7 - √3)/√2 E) (√7- √3)/3 E) √7 - √3

43) which of the following in ascending order if a= √12 + 2, b= √3+ 4, c=  √6+ √8 and d= √2 + √24 ?

A) cbda B) cbad C) cadb D) cabd E) cdab

44) If (x - y)[1/(√x + √y)  + 1/(√x - √y)= 12, find the value of y

A) 36 B) 25 C) 49 D) 64  E) can not be determined

45) If x= 3+ √5, then find the value of x³ - 9x² + 22x. 

A) -15 B) 12 C) 42 D) 45 E) 48

EXERCISE -2

1) If p= (√6 + √3)/(√6 - √3) and q= (√6 - √3)/(√6 + √3), then find the value of p+ q

A) √6 - √3 B) √6 + √3 C) 6 D) 3 E) 4

2) If x= 3+√5, find the value of x² - 1/x².

A) (3- √5)/4 B) 12√5 C) 4 D) (105 + 51 √5)/8 E) (135+ 63 √5)/8

3) If √{31 + 4√21}=  √x + √y and x > y, then the values of x and y respectively are:

A) 27,4 B) 30,1 C) 24,7 D) 21,10 E) 28,3 

4) The positive square root of 41 + 24 √2 is

A) 20+ 2√3 B) 21- 12√2 C) 4√2 - 3 D) 6√2 - 2 E) n

5) Given that x is a rational number and x² - 1= 0, the value of x is

A) (-1- √-3)/2 B) -1 C) (1 + √-3)/2 D) 1 E) n

6) The surd 1/(2+ √3 + √5) is equal to

A) {4+(4√3 - 8)}/16 

B)  {4- (4√3 + 18)}√5/16

C) {4+(4√3 + 8)}√5/16   

D) {4+(4√3 + 18)}√5/16  E( n

7) If √(x + 1/x)= 3, find the value of √(x - 1/x) if x≠ 0

A) 1/3 B) 2/3 C) 1 D) ⁴√79 E) ⁴√77

8) If a, b, c,d are rational √b, √c, √d are irrational √{a+ √b}= √d + c, and a > √b, then find the value of √{a - √b}.

A) c - √d B) - c + √d C) |c - √d| D) √{c - √d{ E) none

9) if a and b are natural numbers and √a, √b are irrational then √(ab) would always be

I. rational if the GCD (a,b) is 1.

II. irrational if the GCD (a,b) is 1.

III. rational if the LCM (a,b) is ab.

IV. irrational if the LCM (a,b) is ab.

A) I and III B) II and IV C) I and IV D) II and III E) III and IV

10) which of the following is the greatest

a) √10 +√7 b) √11 - √8 c)√13 - √10 d) √16 - √14 e) √20 - √17

11)  if a=⁶√(5 + 2√6) and b= ⁶√(5- 2√6), find the value of (a+ b)(a²+ b²+ 2ab -3)= 

A)  2√3 B) 2√2  C) 2√6  D) 10 E) 2√5

12) Evaluate √[5+√{5 - √(5+ √5....

A) (√13 - 1)/2 B) (√17- 1)/2 C) (√17  + 1)/2 D) √17 E) (√13+1)/2

13) if x= 12√6/(√2 + √3), then find the value of (x+ 6√2)/(x - 6√2) + (x + 6 √3)/(x  - 6√3)

A) 2 B) 3 C) 2√3 D) 3√3 E) n

14) which of the following is the Greatest:

a) ₂ √3 +√5 +√6+ √7 

b) ₃√2 +√5 +√6+ √7

c) ₅√2 +√3 +√6+ √7 

d) ₆√2 +√3 +√5+ √7

e) ₇√2 +√3 +√5+ √6

15) simplify: 1/(4+√2) - 1/(4- √2)

A) 2 B)  √6 C) 2/√7 D) -2/√7 E) n

16) Rationalise the denominator 1/(4- √13)

A) (4+√5)/3 B) (4 +√13)/3 C) (4+ √7)/3 D) n

17) Rationalise the denominator 1/(4 + √6 - √10)

A) (4+√5)/3 B) (4 +√13)/3 C) (4+ √7)/3 D) n