Surds Masterclass

What is a Surd?

A surd is a square root that cannot be simplified into a whole number (integer). It is an "irrational number," meaning its decimal form goes on forever without repeating.

In math exams, we leave answers in "surd form" to be exact, rather than rounding a decimal.

1. The Golden Rules of Surds

There are three main laws you need to memorize to manipulate surds.

Rule 1: Multiplication
$$ \sqrt{a} \times \sqrt{b} = \sqrt{a \times b} $$
Example: \(\sqrt{3} \times \sqrt{5} = \sqrt{15}\)
Rule 2: Division
$$ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} $$
Example: \(\frac{\sqrt{20}}{\sqrt{5}} = \sqrt{4} = 2\)
Rule 3: Separation (Simplifying)
$$ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} $$

This is the reverse of Rule 1. We use this to simplify large surds by finding square factors.

Simplify \(\sqrt{50}\)
Think: "What square number goes into 50?" (4, 9, 16, 25...)
Answer: 25.
$$ \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2} $$

2. Adding and Subtracting

You can only add or subtract surds if they are the same type (like algebra).

Think of \(\sqrt{2}\) like \(x\).

3. Expanding Brackets

Treat this exactly like expanding algebraic brackets \((a+b)(c+d)\).

Expand: \((2 + \sqrt{3})(4 - \sqrt{3})\)
1. First: \(2 \times 4 = 8\)
2. Outside: \(2 \times -\sqrt{3} = -2\sqrt{3}\)
3. Inside: \(\sqrt{3} \times 4 = +4\sqrt{3}\)
4. Last: \(\sqrt{3} \times -\sqrt{3} = -\sqrt{9} = -3\)

Collect like terms:
Numbers: \(8 - 3 = 5\)
Surds: \(-2\sqrt{3} + 4\sqrt{3} = +2\sqrt{3}\)
Answer: \(5 + 2\sqrt{3}\)

Practice Questions & Solutions

Try these yourself, then click to reveal the full working out.

Question 1: Simplify \(\sqrt{48}\)

We need the largest square number that goes into 48.

Square numbers: 4, 9, 16, 25...

16 divides into 48 exactly 3 times.

$$ \sqrt{48} = \sqrt{16 \times 3} $$ $$ = \sqrt{16} \times \sqrt{3} $$ $$ = 4 \times \sqrt{3} $$ Answer: \(4\sqrt{3}\)
Question 2: Calculate \(\sqrt{75} + \sqrt{12}\)

We cannot add them yet. We must simplify both first so they have a common surd.

Step 1: Simplify \(\sqrt{75}\)
Largest square factor is 25.
\(\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}\)

Step 2: Simplify \(\sqrt{12}\)
Largest square factor is 4.
\(\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}\)

Step 3: Add them
\(5\sqrt{3} + 2\sqrt{3} = 7\sqrt{3}\)

Answer: \(7\sqrt{3}\)
Question 3: Expand and Simplify \((\sqrt{5} + 2)^2\)

Write it out as double brackets:

$$ (\sqrt{5} + 2)(\sqrt{5} + 2) $$

Use FOIL (First, Outside, Inside, Last):

  • First: \(\sqrt{5} \times \sqrt{5} = 5\)
  • Outside: \(\sqrt{5} \times 2 = 2\sqrt{5}\)
  • Inside: \(2 \times \sqrt{5} = 2\sqrt{5}\)
  • Last: \(2 \times 2 = 4\)

Combine: \(5 + 4 + 2\sqrt{5} + 2\sqrt{5}\)

Answer: \(9 + 4\sqrt{5}\)
Question 4: Simplify \(\frac{18}{\sqrt{3}}\) (Rationalizing the Denominator)

We never leave a surd on the bottom of a fraction. To remove it, multiply the top and bottom by \(\sqrt{3}\).

$$ \frac{18 \times \sqrt{3}}{\sqrt{3} \times \sqrt{3}} $$

The bottom becomes \(\sqrt{9}\), which is just 3.

$$ \frac{18\sqrt{3}}{3} $$

Now simplify the fraction \(18 \div 3 = 6\).

Answer: \(6\sqrt{3}\)
Question 5: Solve for \(x\): \(\sqrt{8} \times \sqrt{10} = x\sqrt{5}\)

Step 1: Combine the left side
\(\sqrt{8} \times \sqrt{10} = \sqrt{80}\)

Step 2: Simplify \(\sqrt{80}\)
Find the largest square factor of 80. It is 16.
\(\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}\)

Step 3: Compare
We have \(4\sqrt{5} = x\sqrt{5}\)

Answer: \(x = 4\)