Surds and Indices: Theory & Concepts

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: 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

Multiplication rule with same base
p ⁿ ⋅ p^m = p^{m + 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
Epample: 3² ⋅ 2² = (3⋅2)² = 36

Division rule with same base
p^m / pⁿ = p^{m - n}
Epample: 3⁵ / 3³ = 3^5-3 = 9
Division rule with same indices
xⁿ / y^m = (x / y)ⁿ
Example: 9³ / 3³ = (9/3)³ = 27

Power rule 1
(aⁿ)^m = a^n.m
Example:
(2³)² = 2^3⋅2 = 2⁶ = 2⋅2⋅2⋅2⋅2⋅2 = 64
Power rule 2
_pn^m = _p(n^m)
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:
27^1/3 = ³√27 = 3

Negative power rule
p^-n = 1 / pⁿ
Example: 2^-2 = 1/2² = 0.25