4 Surds

Definition 4.1. A surd is a root of a rational number that is itself irrational, such as \(\sqrt {2}\), \(\sqrt {3}\), \(\sqrt {5}\) or \(\sqrt [3]{7}\).

The irrationality is part of the definition, and it is what chapter 2 was establishing: \(\sqrt {4}\) is not a surd, because it is \(2\). A surd cannot be written exactly as a fraction or as a terminating decimal, so we keep it in root form and manipulate it symbolically. Writing \(\sqrt {2}\approx 1.414\) throws away exactness for no gain; \(\sqrt {2}\) is already a perfectly good number and every rule below lets us work with it as it stands.

Note that \(\sqrt {2}\) can be written as \(2^{\frac {1}{2}}\), so that \begin {align*} \sqrt {2}\cdot \sqrt {2} & = 2^{\frac {1}{2}}\times 2^{\frac {1}{2}}\\ & = 2^{\frac {1}{2}+\frac {1}{2}}\\ & = 2^1\\ & = 2\\ \end {align*}

Example 4.2.

Express each of the following in its simplest possible surd.

1.: \(\sqrt {12}\)
2.: \(\sqrt {63}\)
3.: \(\sqrt {486}\)

Solution.

1.: \(\sqrt {12}=\sqrt {4\times 3}=\sqrt {4}\times \sqrt {3}=2\sqrt {3}\)
2.: \(\sqrt {63}=\sqrt {9\times 7}=\sqrt {9}\times \sqrt {7}=3\sqrt {7}\)
3.: \(\sqrt {486}=\sqrt {81 \times 6}=\sqrt {81}\times \sqrt {6}=9\sqrt {6}\)

Example 4.3.

In each of the following expand the given expression and simplify.

1.: \((1+2\sqrt {5})(2+\sqrt {5})\)
2.: \((7-3\sqrt {7})(3+2\sqrt {7})\)

Solution.

1.: \begin {align*} (1+2\sqrt {5})(2+\sqrt {5}) &= 1(2+\sqrt {5})+2\sqrt {5}(2+\sqrt {5})\\ & = 2 + \sqrt {5}+4\sqrt {5}+2(5)\\ & = 2 + 10 +5\sqrt {5}\\ & = 12+5\sqrt {5}\\ \end {align*}
2.: \begin {align*} (7-3\sqrt {7})(3+2\sqrt {7}) & = 7(3+2\sqrt {7})-3\sqrt {7}(3+2\sqrt {7})\\ & = 21 + 14\sqrt {7}-9\sqrt {7}-6(7)\\ & = 21-42+5\sqrt {7}\\ & = -21+5\sqrt {7}\\\\ \end {align*}

4.1 Rationalising the Denominator
4.2 Practice Problems

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