##### SURDS AND INDICES

**Important Terms**

**Surd: **Number which cannot be expressed in the fraction form of two integers is called as surd. Hence, the numbers in the form of 3, ^{3}2, ……. ^{n}x

For example: | 1^{1/2} |
can be written as | 1 | but 3 cannot be written in the form of fraction |

9^{1/2} |
3 |

Irrational numbers which contain the radical sign ( ) are called as surds.

**Indices: **Indices refers to the power to which a number is raised. Index is used to show that a number is repeatedly multiplied by itself.

For example: a^{3} is a number with an index of 3 and base ‘a’. It is called as “a to the power of 3”

**Quick Tips and Tricks**

**1) The laws of indices and surds are to be remembered to solve problems on surds and indices. **

**Laws of Indices**

1) x^{m} × x^{n} = a^{m+n}

2) (x^{m})^{n} = x^{mn}

3) (xy) ^{n} = x^{n}y^{n}

4) | x^{m} |
= x^{m – n} |

x^{n} |

5) | x | ^{n} |
= | x^{n} |
||

y | y^{n} |

6) x^{–1} = |
1 |

x |

**Laws of Surds**

1) ^{n}x = x^{(1/n)}

2) ^{n}xy = ^{n}x × ^{n}y

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

^{n}y |

4) (^{n}x)^{n} = X

5) ^{m}^{n}x = ^{mn}x

6) (^{n}x)^{m} = (^{n}x^{m})

**2) Expressing a number in radical form**

Example: l x^{(m/n)} l = ^{n}x^{m}

The exponential form l x^{(m/n)} l is expressed in radical form as ^{n}x^{m}

**Important points to Remember **

**1) **Any number raised to the power zero is always equals to one. (Eg: x ^{0} = 1)

**2) **Surd ^{n}x can be simplified if factor of x is a perfect square

**3) **If denominator in a fraction has any surds, then rationalize the denominator by multiplying both numerator and denominator by a conjugate surd.

**4) **Every surd is an irrational number, but every irrational number is not a surd.

**5) **The conjugate of (2 + 7i) is (2 – 7i)

**6) **Different expressions can be simplified by rationalizing the denominator and eliminating the surd.

**Rationalizing the denominator: **

To rationalize the denominator 7 multiply with its conjugate to both numerator and denominator

Example 1: | 1 | = | 1 | × | 7 | = | 7 |

7 | 7 | 7 | 7 |

Example 2: | 1 | = | 1 | × | 7 – 3 | = | 7 – 3 |

7 + 3 | 7 + 3 | 7 – 3 | 4 |

#### If 2^x × 8^(1/4) = 2^(1/4) then find the value of x ? (^=power)

#### If 9^x – (9)^(x – 1) = 648, then find the value of x^x

#### If 4^ (x – y) = 64 and 4 ^(x + y) = 1024, then find the value of x.

#### If a and b are whole numbers such that a^b = 121, then find the value of (a – 1)^(b + 1)

#### (32)^(n/5) × 2^(2n + 1)/ 4^n × 2^(n – 1)

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