# 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**

3^{1} = 3

3^{2} = 3 × 3 = 9

3^{3} = 3 × 3 × 3 = 27

3^{4} = 3 × 3 × 3 × 3 = 81

###### Surds and Indices rules and properties

**Multiplication rule with same base**

p ^{n} ⋅ p^{m} = p^{m + n}

Example:

2^{3} ⋅ 2^{4} = 2^{3+5} = 2^{8} = 2⋅2⋅2⋅2⋅2⋅2⋅2.2 = 256

**Multiplication rule with same indices**

p^{n} ⋅ y^{n} = (p ⋅ y)^{n}

Example: 3^{2} ⋅ 2^{2} = (3⋅2)^{2} = 36

**Division rule with same base**

p^{m} / p^{n} = p^{m - n}

Example: 3^{5} / 3^{3} = 3^{5-3} = 9

**Division rule with same indices**

x^{n} / y^{m} = (x / y)^{n}

Example: 9^{3} / 3^{3} = (9/3)^{3} = 27

###### Indices power rules

**Power rule 1**

(a

^{n})

^{m}= a

^{n.m}

(2

^{3})

^{2}= 2

^{3⋅2}= 2

^{6}= 2⋅2⋅2⋅2⋅2⋅2 = 64

**Power rule 2**

_{p}n

^{m}=

_{p}(n

^{m})

_{2}3

^{2}=

_{2}(3

^{2}) = 2

^{(3⋅3)}= 2

^{9}= 2⋅2⋅2⋅2⋅2⋅2⋅2⋅2⋅2 = 512

**Power rule 3**

n√p =p1/n

27

^{1/3}=

^{3}√27 = 3

**Negative power rule**

p

^{-n}= 1 / p

^{n}

^{-2}= 1/2

^{2}= 0.25