Surds and Indices  Aptitude basics, practice questions, answers and explanations 
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I. Laws of Indices:

i. a^m * aⁿ = a^m+n

ii. a^m/aⁿ = a^m-n
iii. (a^m)ⁿ =a^mn
iv. (a^b)ⁿ = aⁿbⁿ
v. (a/b)ⁿ = aⁿ/bⁿ
vi. a⁰= 1

II. Surds: Let a be a rational number and n be a positive integer such that a^1/n = n√a is irrational. Then, na is called a surd of order n.

III. Laws of Surds:

1. To find √ (a + √b) write it in the form m + n + 2√mn, such that m + n = a and 4mn = b, then √ (a + √b) = ±(√m + √n)

2. (√a.√a.√a….∞) = a
3. If (√a + √a + √a……..∞) = p, then p (p – 1) = a.
4. If a + √b = c + √d, then a = c and b = d.

Examples:
1. Simplify: (i) (81)^3/4        (ii) (1/64)^-5/6        (iii) (256)^-1/4

Solution:(i) (81)^3/4 =(3⁴)^3/4 =3³=27.
            (ii) (1/64)^-5/6 = 64^5/6 = (2⁶)^5/6= 2⁵ = 32        

            (iii) (256)^-1/4 =( 1/256)^1/4 = [( 1/4)⁴]^1/4 =1/4

2. If x=3+2√2, then the value of ( √x- (1/√x)) is:........
Solution:

Exercise Questions

1.The value of (√8)^1/3 is:
a.2   
b. 4   
c. 2   
d. 8

Answer: Option c.
(√8)^1/3 = (8^1/2)^1/3= 8^1/6 = (2³)^1/6= 2^1/2= √2.

2. The value of 5^1/4 * (125)^0.25 is:
a. √5       
b.5√5       
c.5       
d.25
Answer: Option c
5^0.25 * (5³)^0.25 = 5¹ = 5.

3. The value of (32/243)^-4/5 is:
a. 4/9       
b. 9/4       
c. 16/81       
d. 81/16

Answer: Option d.
(32/243)^-4/5 = (243/32)^4/5 = [(3/2)⁵]^4/5 = 81/16

4. (1/216)^-2/3 ÷ (1/27)^-4/3 = ?
a. 3/4       
b. 2/3       
c. 4/9       
d. 1/8
Answer: Option c.
(1/216)^-2/3 ÷ (1/27)^-4/3 = 216^2/3 ÷ 27^4/3 = (63)^2/3 ÷ (33)^4/3 = 4/9

5. (2ⁿ⁺⁴ -2.2ⁿ)/(2.2ⁿ⁺³) = 2^-3 is equal to:
a. 2ⁿ⁺¹       
b. -2ⁿ⁺¹ + 1/8        
c. 9/8 - 2ⁿ       
d. 1
Answer: Option d.

(2ⁿ⁺⁴ -2.2ⁿ)/2.2ⁿ⁺³  + 1/2³= 7/8 + 1/8 = 1  

6. If 5√5 * 5³ ÷ 5^-3/2 = 5^a+2 , the value of a is:
a. 4       
b. 5       
c.6       
d. 8

Answer: Option a
5^3/2 * 5³ ÷ 5^-3/2 = 5^a+2

5^3/2 + 3 + 3/2 = 5^a+2
3/2 + 3 + 3/2 = a+2
a+2=6; a=4

7. If √2n =64, then the value of n is:
a. 2       
b. 4       
c. 6       
d. 12

Answer: Option d
√2n =64 => 2^n/2 = 64= 2⁶
n/2=6; n=12

8.The simplified form of (x^7/2 /x^5/2).√y3 /√y )is :
a.x²/y       
b. x³/y²           
c. x⁶/y³       
d. xy
Answer: Option d

(x^7/2 /x^5/2 ). (√y3 /√y) = x^7/2 -^5/2. y^3/2 - ^1/2 = xy