Surds & Indices Practice Set - Fully Solved

Q 1.(27)²/₃=x then 3x is equal to
a) 6 b) 3 c) 27 d)9 e) None  

Q 2. ( ²⁷/₁₂₅ )^-2/3 = ? 
a) 625/18 b) 25/9 c) 81/625 e) 9/25

Q 3. If (64)^2/3 × (256)^-1/2 = 4ⁿ , then n= ?
a) 2 b) 4 c) 0 d) 1 e) None

Q 4.(64)^0.25 × (36)^1.5= ?
a) None b) 108 c) 48 d) 54 e) 1021

Q 5. ^3√1000000/_6√1000000=(100)^x , then x is equal to .
a) 4 b) 1/2 c) None of these d) 1/4 e) 2

Q 6. (-¹/₁₃₃₁)^-2/3× √1331 × (¹/₁₂₁)^3/2 
a) (¹/₁₁)^-1/2 b) (-¹/₁₁)^1/3 c) (-¹/₁₁)^2/3 d) (-¹/₁₂₁)^2/3 e) 11³/₂

Q7.(⁶⁴/₇₂₉)^-1/3 = ?
a) -⁴/₉ b) ⁸/9 c) ⁸/27 d) 2¹/₄ e) None of these

Q 8. Simplify [x^2n-1 + y^2n-1]²[x^2n-1 - y^2n-1] .
a) x²ⁿ+ y²ⁿ b) x²ⁿ c) -y²ⁿ d) x²ⁿ- y²ⁿ e) None of these

Q 9. Arrange 2^{2-1 , 4 0.33 , 60.25} in ascending order .
a)1st<2nd<3 b)3rd<2nd<1 c)1st<3rd<2 d)3rd<1st<2 e) None of these

Q 10. Evaluate (0.04)^-1.5× (0.125)^-4/3- (¹/₁₂₁)^-1/2 
a) 1989 b) 22000 c) 2045 d) 2011 e) 2000 ¹/₁₁

Q 11. Express ²/₃ √32 as a pure surd .
a)√128 b)√128/9 c)√9/128 d) e) None of these

Solution

Sol: Option (c)
(27)^2/3 = x ⇒ (3³)^2/3 = x ⇒ 3² = x ⇒ 3x = 2

(2)
Sol: Option (b)
(²⁷/₁₂₅)^-2/3 = (¹²⁵/₂₇)^2/3 = ²⁵/9

(3)

Sol :Option (c)
(4³)^2/3×(4⁴)^-1/2 = 4ⁿ
4² × 4^-2 = 4ⁿ, ⇒4⁰⇒ n = 0

(4)
Sol : Option (a)
(√64²)^1/3=(64)^1/3=(4³)^1/3 = 4

(5)
Sol : Option (b)
^3√1003/_6√100³ = (100)^x ⇒ 100¹/₂ = (100)^x⇒ x = ¹/₂

(6) 
Sol: Option (a)
Convert each term to the base of 11
(-¹/₁₃₃₁)^-2/3 = (-1331)^2/3 = (-11)^3*2/₃ = (-11)² = 11²
√1331 = 11^{3/2 , (1/₁₂₁)3/2 = 11-2*3/₂ = 11-3}

(7)
Sol: Option (d)
^{(43/₉₃)-1/3
= (4/₉)-1
= 9/_{4
= 21/4}}

(8)
Sol : Option (d)
^{Given expression = ( a+b ) ( a-b ) = a2 - b2
= [x2n-1 ]2- [y2n-1]2
= x2n-1×21 - y2n-1×21
= x2n- y2n}

(9)
Sol : Option (c)

(2^1/2) , (4 ^1/3) , (6 ^1/4) 

Here, the base and the index both are different .For comparision, either the base should be same or indices of each number should be same. Take LCM of 2,3,4 = 12 and convert each number to same index
(2^1/2) = (2⁶)^1/12 ; (4⁴)^1/12 ; (6³)^1/4
= (64)^1/12 = (256)^1/12 = (216)^1/12
Clearly,(64)^1/12 < (216)^1/12 < (256)^1/12  ⇒2^1/2< 6^0.25< 4^0.33

(10)
Sol : Option (a)

Given expression = (0.2)^-3×(0.5)^-4 - 11 = (0.2)^-3×(0.5)^-3.(0.5)^-1-11

= (0.1)^-3×(0.5)^-1-11 = 2000-11

(11)
Sol : Option (b)

²/₃ √32 = √(²/₃)²×32
=√¹²⁸/₉ a pure surd

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