The base x raised to the power of p is equal to the multiplication of x, p timesx = x × x × ... × x p times. x is the base and p is the indices. Surds and indices maths problems have a frequent appearance in some of the entrance exams.

Examples
3¹ = 3
3² = 3 × 3 = 9
3³ = 3 × 3 × 3 = 27
3⁴ = 3 × 3 × 3 × 3 = 81

Numbers which can be expressed in the form √p + √q , where p and q are natural numbers and not perfect squares. Irrational numbers which contain the radical sign (n√) are called as surds Hence, the numbers in the form of √3, 3√2, ……. n√x

Examples:

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

Solution:
(√8)^1/3 = (8^1/2)^1/3= 8^1/6 = (2³)^1/6= 2^1/2= √2.

Question 2: The value of 5^1/4 * (125)^0.25 is:
a. √5              b.5√5                      c.5                       d.25

Solution: 
5^0.25 * (5³)^0.25 = 5¹ = 5.

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

Solution:
(32/243)^-4/5 = (243/32)^4/5 = [(3/2)⁵]^4/5 = 81/16

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

Solution:
(1/216)^-2/3 ÷ (1/27)^-4/3 = 216^2/3 ÷ 27^4/3 = (63)^2/3 ÷ (33)^4/3 = 4/9

Question 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

Solution:

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

Question 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

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

Solution:
√2ⁿ =64 => 2^n/2 = 64= 2⁶
n/2=6; n=12

Question 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

Solution:

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