**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:** 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 and Surds rules and properties

**Rule name** |
**Rule** |

Multiplication Rule |
p^{n}⋅ p^{m} = p^{m+n} |

p^{n} ⋅ q^{n} = (p ⋅ q)^{n} |

Division Rule |
p^{m}/ p^{n} = x^{m-n} |

p^{n} / q^{n} = (p / q)^{n} |

Power Rule |
(p^{n})^{m} = p^{n⋅m} |

_{p}n^{m} = _{p}(n^{m}) |

^{m}√(p^{n}) = p ^{n/m} |

^{n}√p = p^{1/n} |

p^{-n} = 1 / p^{n} |

###### Indices Multiplication rules

**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}

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

Must Read Surds and Indices Articles

###### Indices division rules

**Division rule with same base**

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

Epample: 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}

Example:

(2^{3})^{2} = 2^{3⋅2} = 2^{6} = 2⋅2⋅2⋅2⋅2⋅2 = 64

**Power rule 2**

_{p}n^{m} = _{p}(n^{m})

Example:

_{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

Example:

27^{1/3} = ^{3}√27 = 3

**Negative power rule**

p^{-n} = 1 / p^{n}

Example: 2^{-2} = 1/2^{2} = 0.25