Explanations
LEVEL 1 – BASIC INDICES

Solutions 1–20

1.

2³ × 2⁴
= 2^(3+4)
= 2⁷

Answer: A

2.

5⁷ ÷ 5³
= 5^(7-3)
= 5⁴

Answer: B

3.

Using a⁰ = 1,

3⁰ = 1

Answer: B

4.

(2³)⁴
= 2^(3×4)
= 2¹²

Answer: B

5.

2⁻³
= 1/2³
= 1/8

Answer: C

6.

7⁸/7⁵
= 7^(8-5)
= 7³

Answer: A

7.

3⁴ × 3⁻²
= 3^(4-2)
= 3²

Answer: B

8.

16^(1/2)
= √16
= 4

Answer: B

9.

27^(1/3)
= ∛27
= 3

Answer: A

10.

32^(1/5)
= ⁵√32
= 2

Answer: A

11.

(2⁵ × 2³)/2⁴
= 2^(5+3-4)
= 2⁴

Answer: B

12.

2ˣ = 32

32 = 2⁵

Therefore,

x = 5

Answer: C

13.

3ˣ = 81

81 = 3⁴

Therefore,

x = 4

Answer: C

14.

(5²)³ ÷ 5⁴
= 5⁶ ÷ 5⁴
= 5²

Answer: A

15.

81^(1/2)
= √81
= 9

Answer: B

16.

64^(1/3)
= ∛64
= 4

Answer: B

17.

(2/3)⁻²
= (3/2)²
= 9/4

Answer: B

18.

4⁵/2⁸

Since 4 = 2²,

= (2²)⁵/2⁸
= 2¹⁰/2⁸
= 2²
= 4

Answer: B

19.

5^(x+1) = 125

125 = 5³

Therefore,

x + 1 = 3

x = 2

Answer: B

20.

3^(x+4)/3^(x+1)

= 3^[(x+4)-(x+1)]

= 3³

= 27

Answer: C

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LEVEL 2 – BASIC SURDS

Solutions 21–40

21.

√12
= √(4×3)
= 2√3

Answer: A

22.

√18
= √(9×2)
= 3√2

Answer: B

23.

√50
= √(25×2)
= 5√2

Answer: B

24.

√72
= √(36×2)
= 6√2

Answer: B

25.

√98
= √(49×2)
= 7√2

Answer: A

26.

√48
= √(16×3)
= 4√3

Answer: C

27.

√75
= √(25×3)
= 5√3

Answer: B

28.

2√3 + 5√3
= (2+5)√3
= 7√3

Answer: A

29.

8√5 - 3√5
= (8-3)√5
= 5√5

Answer: A

30.

√12 + √27

= 2√3 + 3√3

= 5√3

Answer: C

31.

√32 + √8

= 4√2 + 2√2

= 6√2

Answer: C

32.

√45 - √5

= 3√5 - √5

= 2√5

Answer: B

33.

√3 × √12

= √36

= 6

Answer: B

34.

√8 × √2

= √16

= 4

Answer: B

35.

(2√3)(3√6)

= 6√18

= 6×3√2

= 18√2

Answer: C

36.

√50/√2

= √(50/2)

= √25

= 5

Answer: B

37.

√108/√3

= √36

= 6

Answer: B

38.

(√7)² = 7

Answer: B

39.

(√5 + √3)(√5 - √3)

Using (a+b)(a-b) = a²-b²,

= 5 - 3

= 2

Answer: A

40.

(√3 + √2)²

= 3 + 2 + 2√6

= 5 + 2√6

Answer: B

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LEVEL 3 – FRACTIONAL INDICES AND MIXED SURDS

Solutions 41–60

41.

16^(3/4)

= (⁴√16)³

= 2³

= 8

Answer: C

42.

27^(2/3)

= (∛27)²

= 3²

= 9

Answer: C

43.

32^(2/5)

= (⁵√32)²

= 2²

= 4

Answer: B

44.

81^(-1/2)

= 1/81^(1/2)

= 1/9

Answer: B

45.

8^(-2/3)

= 1/8^(2/3)

= 1/(∛8)²

= 1/2²

= 1/4

Answer: B

46.

[2^(5/2) × 2^(3/2)]/2²

= 2^(5/2+3/2-2)

= 2^(8/2-2)

= 2²

= 4

Answer: B

47.

3^(7/2)/3^(3/2)

= 3^[(7-3)/2]

= 3²

= 9

Answer: C

48.

2^(x+1) = 8^(x-1)

8 = 2³

2^(x+1) = 2^(3x-3)

Therefore,

x + 1 = 3x - 3

2x = 4

x = 2

Answer: B

49.

9ˣ = 27

(3²)ˣ = 3³

3^(2x) = 3³

2x = 3

x = 3/2

Answer: C

50.

√200 - √72

= 10√2 - 6√2

= 4√2

Answer: B

51.

3√12 + 2√27 - √75

= 3(2√3) + 2(3√3) - 5√3

= 6√3 + 6√3 - 5√3

= 7√3

Answer: C

52.

(√5 + √2)² - (√5 - √2)²

Using:

(a+b)² - (a-b)² = 4ab

= 4√5√2

= 4√10

Answer: B

53.

(√7 + √3)(√7 - √3)

= 7 - 3

= 4

Answer: B

54.

1/(√5 + √3)

Multiply by the conjugate:

= (√5 - √3)/(5-3)

= (√5 - √3)/2

Answer: A

55.

1/(√3+√2) + 1/(√3-√2)

Since:

1/(√3+√2) = √3-√2

and

1/(√3-√2) = √3+√2

Therefore,

= 2√3

Answer: B

56.

1/(√5-√3) - 1/(√5+√3)

= (√5+√3)/2 - (√5-√3)/2

= 2√3/2

= √3

Answer: A

57.

x = √3 + √2

1/x = √3 - √2

Therefore,

x + 1/x

= 2√3

Answer: B

58.

x + 1/x = 2√3

Squaring,

x² + 1/x² + 2 = 12

x² + 1/x² = 10

Answer: C

59.

Let:

a = √2 + √3
b = √2 - √3

Using:

(a+b)³ type direct expansion,

(√2+√3)³ + (√2-√3)³

= 2(√2)³ + 6(√2)(√3)²

= 4√2 + 18√2

= 22√2

Answer: D

60.

x = √5 + √2

Since:

(√5+√2)(√5-√2) = 3

Therefore,

3/x = √5 - √2

So,

x + 3/x = 2√5

Squaring,

x² + 9/x² + 6 = 20

x² + 9/x² = 14

Answer: B

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LEVEL 4 – RATIONALISATION AND ADVANCED SURDS

Solutions 61–80

61.

1/√5

Multiply numerator and denominator by √5:

= √5/5

Answer: B

62.

3/√6

= 3√6/6

= √6/2

Answer: A

63.

1/(√7 + √6)

Multiply by √7-√6:

= (√7-√6)/(7-6)

= √7-√6

Answer: B

64.

1/(√7-√5)

= (√7+√5)/(7-5)

= (√7+√5)/2

Answer: A

65.

(√6+√5)/(√6-√5)

Multiply by √6+√5:

= (√6+√5)²/(6-5)

= 6+5+2√30

= 11+2√30

Answer: A

66.

(√5+1)/(√5-1)

Multiply by √5+1:

= (√5+1)²/(5-1)

= (6+2√5)/4

= (3+√5)/2

Answer: A

67.

1/(3+2√2)

Since:

(3+2√2)(3-2√2)

= 9-8

= 1

Therefore,

1/(3+2√2) = 3-2√2

Answer: B

68.

√(9+4√5)

Assume:

√(9+4√5) = √a + √b

Then:

a+b = 9
2√ab = 4√5

ab = 20

So a = 5 and b = 4.

Therefore,

= √5 + 2

Answer: A

69.

√(7+4√3)

= √4 + √3

= 2+√3

because:

(2+√3)² = 7+4√3

Answer: A

70.

√(7-4√3)

= 2-√3

because:

(2-√3)² = 7-4√3

Answer: B

71.

√(5+2√6)

= √3 + √2

because:

(√3+√2)²

= 3+2+2√6

= 5+2√6

Answer: A

72.

√(5-2√6)

= √3 - √2

Answer: B

73.

x = √2+1

1/x

= 1/(√2+1)

= √2-1

Answer: B

74.

x = √2+1

1/x = √2-1

Therefore,

x + 1/x = 2√2

Squaring,

x² + 1/x² + 2 = 8

x² + 1/x² = 6

Answer: C

75.

x = √3+√2

x + 1/x = 2√3

Therefore,

x² + 1/x²

= (2√3)² - 2

= 12-2

= 10

Answer: B

76.

Let x = √3+√2.

Then:

1/x = √3-√2

We know:

x² + 1/x² = 10

Therefore,

x⁴ + 1/x⁴

= (x²+1/x²)² - 2

= 10² - 2

= 98

Answer: C

77.

x + 1/x = 2√3

Using:

x³ + 1/x³

= (x+1/x)³ - 3(x+1/x)

= (2√3)³ - 6√3

= 24√3 - 6√3

= 18√3

Answer: C

78.

√(12+8√2)

Observe:

(2√2+2)²

= 8+4+8√2

= 12+8√2

Therefore,

√(12+8√2) = 2√2+2

Answer: B

79.

√(12-8√2)

= 2√2-2

because:

(2√2-2)² = 12-8√2

Answer: A

80.

a = √5+2

Since:

(√5+2)(√5-2)=1

Therefore,

1/a = √5-2

So,

a + 1/a

= 2√5

Answer: A

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LEVEL 5 – TOUGH SSC & RAILWAY LEVEL

Solutions 81–100

81.

x = 2+√3

Since:

1/x = 2-√3

Therefore,

x + 1/x = 4

So,

x² + 1/x²

= 4² - 2

= 14

Answer: B

82.

x + 1/x = 4

Using:

x³ + 1/x³

= (x+1/x)³ - 3(x+1/x)

= 64 - 12

= 52

Answer: C

83.

x = √5+√3

Since:

(√5+√3)(√5-√3)=2

Therefore,

2/x = √5-√3

So:

x + 2/x = 2√5

Squaring:

x² + 4/x² + 4 = 20

Therefore,

x² + 4/x² = 16

Answer: C

84.

Let:

a = √7+√5
b = √7-√5

We need:

a/b + b/a

= (a²+b²)/ab

Now:

ab = 7-5 = 2

a²+b²

= 2(7+5)

= 24

Therefore,

= 24/2

= 12

Answer: C

85.

x = √3+√2

We know:

x² + 1/x² = 10

Therefore,

x⁴ + 1/x⁴

= 10² - 2

= 98

Answer: C

86.

x + 1/x = 3

Squaring:

x² + 1/x² + 2 = 9

Therefore,

x² + 1/x² = 7

Answer: B

87.

x + 1/x = 4

x³ + 1/x³

= 4³ - 3(4)

= 64-12

= 52

Answer: B

88.

x - 1/x = 2

Squaring:

x² + 1/x² - 2 = 4

Therefore,

x² + 1/x² = 6

Answer: B

89.

x - 1/x = √5

Squaring:

x² + 1/x² - 2 = 5

Therefore,

x² + 1/x² = 7

Answer: C

90.

(√5+√3)² + (√5-√3)²

Using:

(a+b)²+(a-b)² = 2(a²+b²)

= 2(5+3)

= 16

Answer: C

91.

(√7+√2)² - (√7-√2)²

Using:

(a+b)²-(a-b)² = 4ab

= 4√14

Answer: B

92.

Let:

x = √3+√2

Then:

1/x = √3-√2

x + 1/x = 2√3

First:

x³ + 1/x³

= (2√3)³ - 3(2√3)

= 18√3

Therefore,

x⁶ + 1/x⁶

= (18√3)² - 2

= 972 - 2

= 970

Answer: A

93.

1/(√2-1)²

Since:

1/(√2-1) = √2+1

Therefore,

= (√2+1)²

= 3+2√2

Answer: A

94.

1/(2-√3)³

Since:

1/(2-√3) = 2+√3

Therefore,

= (2+√3)³

Now:

(2+√3)² = 7+4√3

Multiplying by 2+√3:

= 14+7√3+8√3+12

= 26+15√3

Answer: A

95.

4ˣ = 8^(x-1)

2^(2x) = 2^(3x-3)

Therefore,

2x = 3x-3

x = 3

Answer: B

96.

27^(x+1) = 9^(2x-1)

3^(3x+3) = 3^(4x-2)

Therefore,

3x+3 = 4x-2

x = 5

Answer: C

97.

2^(2x) - 5(2ˣ) + 4 = 0

Let:

y = 2ˣ

Then:

y² - 5y + 4 = 0

(y-1)(y-4)=0

So:

y = 1 or 4

Therefore:

2ˣ = 1 ⇒ x=0

or

2ˣ = 4 ⇒ x=2

Hence:

x = 0 or 2

Answer: B

98.

3ˣ + 3^(x+1) = 108

3ˣ + 3×3ˣ = 108

4×3ˣ = 108

3ˣ = 27

3ˣ = 3³

Therefore,

x = 3

Answer: B

99.

(81/16)^(-3/4)

81/16 = (3/2)⁴

Therefore,

[(3/2)⁴]^(-3/4)

= (3/2)⁻³

= (2/3)³

= 8/27

Answer: A

100.

[16^(3/4) × 8^(2/3)]/32^(1/5)

16^(3/4) = 8

8^(2/3) = 4

32^(1/5) = 2

Therefore,

= (8×4)/2

= 16

Answer: C

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LEVEL 6 – TOUGHEST / CONCEPT-TRAP LEVEL

Solutions 101–120

101.

x = √2+√3

Since:

1/x = √3-√2

Therefore:

x + 1/x = 2√3

Let:

S₁ = x+1/x = 2√3

S₂ = S₁²-2
= 12-2
= 10

S₃ = S₁S₂-S₁
= 20√3-2√3
= 18√3

S₄ = S₁S₃-S₂
= 108-10
= 98

S₅ = S₁S₄-S₃
= 196√3-18√3
= 178√3

Therefore,

x⁵ + 1/x⁵ = 178√3

Correct Answer: 178√3

Correction: None of the options in the earlier Q101 are correct. Replace the correct option with 178√3.

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102.

x + 1/x = 5

First:

x² + 1/x²

= 25-2

= 23

Now:

x⁴ + 1/x⁴

= 23²-2

= 529-2

= 527

Answer: A

103.

x - 1/x = 3

Squaring:

x² + 1/x² - 2 = 9

So:

x² + 1/x² = 11

Therefore:

x⁴ + 1/x⁴

= 11²-2

= 119

Answer: A

104.

x + 1/x = √5

We use recurrence.

x² + 1/x²

= 5-2

= 3

x³ + 1/x³

= √5×3 - √5

= 2√5

x⁴ + 1/x⁴

= √5(2√5)-3

= 10-3

= 7

x⁵ + 1/x⁵

= √5×7 - 2√5

= 5√5

Answer: A

105.

x = 2+√3

1/x = 2-√3

Therefore:

x + 1/x = 4

So:

x² + 1/x²

= 16-2

= 14

Hence:

x⁴ + 1/x⁴

= 14²-2

= 194

Answer: C

106.

√(19+6√10)

Assume:

= √a + √b

Then:

a+b = 19

2√ab = 6√10

ab = 90

The numbers are 9 and 10.

Therefore:

√(19+6√10)

= 3+√10

Answer: A

107.

√(19-6√10)

= √10-3

because:

(√10-3)²

= 10+9-6√10

= 19-6√10

Answer: B

108.

√(13+4√10)

Observe:

(√5+2√2)²

= 5+8+4√10

= 13+4√10

Therefore:

√(13+4√10)

= √5+2√2

Answer: C

Note: In the earlier options, option C was written as 2√2 + √5, which is the same value.

109.

√(13-4√10)

Observe:

(√5-2√2)²

= 5+8-4√10

= 13-4√10

Since 2√2 > √5,

principal square root is:

2√2-√5

Answer: D

Correction: The earlier answer marked B was incorrect. The correct answer is D. 2√2 - √5.

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110.

a = √3+√2
b = √3-√2

ab = 1

a²+b²

= 2(3+2)

= 10

Therefore:

a⁴+b⁴

= (a²+b²)² - 2a²b²

= 100-2

= 98

Answer: C

111.

a = 2+√3
b = 2-√3

a+b = 4
ab = 1

Let:

S₀ = 2
S₁ = 4

Using:

Sₙ = 4Sₙ₋₁ - Sₙ₋₂

S₂ = 4×4-2 = 14

S₃ = 4×14-4 = 52

S₄ = 4×52-14 = 194

S₅ = 4×194-52

= 776-52

= 724

Therefore:

a⁵+b⁵ = 724

Answer: A

112.

x = √5+2

Since:

1/x = √5-2

Therefore:

x + 1/x = 2√5

Using:

x³ + 1/x³

= (x+1/x)³ - 3(x+1/x)

= (2√5)³ - 6√5

= 40√5 - 6√5

= 34√5

Correct Answer: 34√5

Correction: The earlier options and answer for Q112 were incorrect. The correct value is 34√5.

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113.

x = √5+2

x + 1/x = 2√5

Therefore:

x² + 1/x²

= (2√5)² - 2

= 20-2

= 18

Hence:

x⁴ + 1/x⁴

= 18²-2

= 324-2

= 322

Correct Answer: 322

Correction: The earlier options and answer were incorrect.

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114.

We have:

1/(√5+2) = √5-2

and:

1/(√5-2) = √5+2

Therefore:

1/(√5+2)⁴ + 1/(√5-2)⁴

= (√5-2)⁴ + (√5+2)⁴

From Q113:

= 322

Correct Answer: 322

Correction: The earlier options and answer were incorrect.

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115.

x = √3+√2

1/x = √3-√2

Therefore:

x + 1/x = 2√3

We know:

x³ + 1/x³ = 18√3

Hence:

x⁶ + 1/x⁶

= (18√3)² - 2

= 324×3 - 2

= 972-2

= 970

Answer: C

116.

2ˣ = 3

Therefore:

2^(2x) = (2ˣ)² = 9

and:

2^(-2x) = 1/9

Hence:

2^(2x) + 2^(-2x)

= 9 + 1/9

= 82/9

Answer: B

117.

3ˣ = 2

9ˣ

= (3²)ˣ

= (3ˣ)²

= 4

Also:

9⁻ˣ = 1/4

Therefore:

9ˣ + 9⁻ˣ

= 4 + 1/4

= 17/4

Answer: C

118.

5ˣ = 3

25ˣ

= (5²)ˣ

= (5ˣ)²

= 9

Therefore:

25⁻ˣ = 1/9

Hence:

25ˣ + 25⁻ˣ

= 9 + 1/9

= 82/9

Answer: B

119.

x + 1/x = 6

Let Sₙ = xⁿ + 1/xⁿ.

S₀ = 2
S₁ = 6

S₂ = 6×6-2
= 34

S₃ = 6×34-6
= 198

S₄ = 6×198-34
= 1154

S₅ = 6×1154-198
= 6924-198
= 6726

Therefore:

x⁵ + 1/x⁵ = 6726

Answer: A

120.

x - 1/x = 1

Squaring:

x² + 1/x² - 2 = 1

Therefore:

x² + 1/x² = 3

Now:

x³ - 1/x³

= (x-1/x)[x² + 1/x² + 1]

= 1(3+1)

= 4

Squaring:

(x³ - 1/x³)² = 16

x⁶ + 1/x⁶ - 2 = 16

Therefore:

x⁶ + 1/x⁶ = 18

Answer: B