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surd is a root (e.g. \(\sqrt{n}\), \(\sqrt{3}\), \(\sqrt{17}\) and \(7\sqrt{2}\)). Surds are
irrational numbers because they cannot be expressed as \(a/b\). They are non-recurring decimals, e.g. \(\sqrt{3}=1.732050807568877...\)
Addition: You can only add surds if the roots are the same, or can be simplified such that they become the same. Otherwise, you cannot add them.
Subtraction: Same as above.
Multiplication: \(\sqrt{a}×\sqrt{b}=\sqrt{ab}\)
Division: \(\dfrac{\sqrt{a}}{\sqrt{b}}=\sqrt{\dfrac{a}{b}}\)
Rationalising the denominator
If a fraction has a surd in the denominator, you can rewrite it such that the denominator is a
rational number. This is known as rationalising the denominator.
This can be done by using the following results:
\((\sqrt{x})^2=x\) and
\((\sqrt{x}+\sqrt{y})(\sqrt{x}-\sqrt{y})=x-y\)
Rules of surds:
- \(\sqrt{a}×\sqrt{b}=\sqrt{ab}\)
- \(\dfrac{\sqrt{a}}{\sqrt{b}}=\sqrt{\dfrac{a}{b}}\)
Rationalising the denominator:
- For fractions of the form \(\dfrac{n}{\sqrt{a}}\), multiple by \(\dfrac{\sqrt{a}}{\sqrt{a}}\).
- For fractions of the form \(\dfrac{n}{a±\sqrt{b}}\), multiple by \(\dfrac{a∓\sqrt{b}}{a∓\sqrt{b}}\).
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