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This document was AI generated.

Surds are numbers left in square root (or cube root) form because their exact value cannot be written as a whole number or a fraction. They are irrational numbers, meaning their decimals go on forever without repeating.

For example, $\sqrt{4} = 2$ (not a surd), but $\sqrt{2} = 1.4142…$ (is a surd).

Core Rules

There are two main rules used to manipulate and simplify surds:

• Multiplication: $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$
• Division: $\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}$

Note

You cannot split addition or subtraction using this. $\sqrt{a + b}$ is not equal to $\sqrt{a} + \sqrt{b}$.

Simplifying

To simplify a surd, find the largest square number (like 4, 9, 16, 25, 36…) that divides into the number under the root.

Example ($\sqrt{50}$)

1. Find a square number that goes into 50. That’s 25.
2. Rewrite it: $\sqrt{50} = \sqrt{25 \times 2}$
3. Split it up: $\sqrt{25} \times \sqrt{2}$
4. Simplify the square root: $5\sqrt{2}$

Adding and Subtracting

You can only add or subtract “like” surds, just like collecting like terms in algebra ($x + 2x = 3x$).

Example ($\sqrt{12} + \sqrt{27}$)

At first glance, they don’t match. You need to simplify them first.

Now add them together:

$2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}$

Rationalising the Denominator

Math convention says we shouldn’t leave a root on the bottom (denominator) of a fraction. “Rationalising” means getting rid of the root on the bottom.

Case 1: Single surd on the bottom

Multiply the top and bottom by that surd.

• Example: Rationalise $\frac{5}{\sqrt{3}}$

$\frac{5 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} = \frac{5\sqrt{3}}{3}$

Case 2: A mix of numbers and surds on the bottom

Multiply the top and bottom by the denominator, but change the sign in the middle (this uses the difference of two squares to cancel out the middle root terms).

• Example: Rationalise $\frac{2}{3 + \sqrt{5}}$
  1. Multiply top and bottom by $(3 - \sqrt{5})$:

$\frac{2(3 - \sqrt{5})}{(3 + \sqrt{5})(3 - \sqrt{5})}$

2. Expand the bottom: $(3 \times 3) - (3\sqrt{5}) + (3\sqrt{5}) - (\sqrt{5} \times \sqrt{5}) = 9 - 5 = 4$
3. Put it back together:

$\frac{2(3 - \sqrt{5})}{4} = \frac{3 - \sqrt{5}}{2}$