2.2 Surds

AQA Edexcel OCR A OCR B (MEI)
A 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

\((\sqrt{x}+\sqrt{y})(\sqrt{x}-\sqrt{y})=x-y\)
Important
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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