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
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Indices and Surds rules and properties
Rule name Rule
Multiplication Rule pn⋅ pm = pm+n
pn ⋅ qn = (p ⋅ q)n
Division Rule pm/ pn = xm-n
pn / qn = (p / q)n
Power Rule (pn)m = pn⋅m
pnm = p(nm)
m√(pn) = p n/m
n√p = p1/n
p-n = 1 / pn
Indices Multiplication rules
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
Epample: 32 ⋅ 22 = (3⋅2)2 = 36
Indices division rules
Division rule with same base
pm / pn = pm - n
Epample: 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
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