Surds and Indices

Surds: 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

Indices: 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

Surds and Indices rules and properties

Multiplication rule with same base
p n ⋅ pm = pm + n

Example:
23 ⋅ 24 = 23+5 = 28 = 2⋅2⋅2⋅2⋅2⋅2⋅2.2 = 256

Multiplication rule with same indices
pn ⋅ yn = (p ⋅ y)n

Example: 32 ⋅ 22 = (3⋅2)2 = 36

Division rule with same base
pm / pn = pm - n

Example: 35 / 33 = 35-3 = 9

Division rule with same indices
xn / ym = (x / y)n

Example: 93 / 33 = (9/3)3 = 27

Indices power rules
Power rule 1
(an)m = an.m
 
Example:
(23)2 = 23⋅2 = 26 = 2⋅2⋅2⋅2⋅2⋅2 = 64
 
Power rule 2
pnm = p(nm)
 
Example:
232 = 2(32) = 2(3⋅3) = 29 = 2⋅2⋅2⋅2⋅2⋅2⋅2⋅2⋅2 = 512
 
Power rule 3
n√p =p1/n
 
Example:
271/3 = 3√27 = 3
 
Negative power rule
p-n = 1 / pn
 
Example: 2-2 = 1/22 = 0.25