A surd is a root that is irrational.

Examples of surds include √a, √2, 3√11, √a + √2, 3√15

Use the following rules to simplify surds:

√ab = √a x √b

√a/√b = √(a/b)

Remember that √a + b ≠ √a + √b

**Example
**

Simplify the following as a surd in the form a + b√c, where c is an integer and is as small as possible, or if the answer is a whole number, just state the whole number.

(3 + √5)(3 - 2√5) |

√45 |

**Answer**

First we simplify the numerator remembering that √5 x √5 = 5

(3 + √5)(3 - 2√5)

= 3(3 - 2√5) + √5(3 - 2√5)

= 9 - 6√5 + 3 - 2 x 5

= 9 + 3 - 10 - 6√5

= 2 - 6√5

So we now have

2 - 6√5 |

√45 |

which is equal to

2 - 6√5 |

√9 x 5 |

which is equal to

2 - 6√5 |

3√5 |

multiply top and bottom by √5 to get

2√5 - 6 x 5 |

3 x 5 |

which simplifies to

2√5 - 30 |

15 |

which equals

2√5 |
- 2 |

15 |