Surd

A surd is an irrational number that cannot be simplified to remove the square root (or cube root, etc.) and still remain exact. It is usually left in root form (not as a decimal).

• \( \sqrt{2} \), \( \sqrt{3} \), \( \sqrt{5} \) are examples of surds — they are irrational and non-terminating.
• \( \sqrt{4} = 2 \) is not a surd because it simplifies to a rational number.

Surds are used to give exact answers without rounding. They maintain accuracy in geometry (e.g. Pythagoras’ Theorem) and algebra.

Rules of Surds:

• \( \sqrt{a} \times \sqrt{b} = \sqrt{ab} \)
• \( \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \)
• \( (\sqrt{a})^2 = a \)
• \( \sqrt{a^2} = a \), if \( a \geq 0 \)

Simplifying Surds:

Break down the number inside the square root into factors, one of which is a perfect square.

• \( \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2} \)
• \( \sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2} \)

Adding & Subtracting Surds:

You can only add or subtract surds if they have the same radical part (just like like terms in algebra).

• \( 3\sqrt{2} + 5\sqrt{2} = 8\sqrt{2} \)
• \( 4\sqrt{3} – 2\sqrt{3} = 2\sqrt{3} \)
• \( 2\sqrt{5} + 3\sqrt{2} \) cannot be simplified further

Multiplying & Dividing Surds:

• \( \sqrt{3} \times \sqrt{12} = \sqrt{36} = 6 \)
• \( \frac{\sqrt{18}}{\sqrt{2}} = \sqrt{9} = 3 \)