Manipulating Surds Revision Notes for AQA GCSE Maths

Manipulating surds

What are surds?

Surds are mathematical expressions that contain irrational square roots - these are square roots that cannot be written as simple fractions or whole numbers. Examples include most square roots like $\sqrt{2}$ , $\sqrt{3}$ , $\sqrt{5}$ , and cube roots like $\sqrt[3]{7}$ . You'll recognise them because they can't be simplified to give you a neat decimal or fraction.

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Common examples of surds include $\sqrt{2}$ , $\sqrt{3}$ , $\sqrt{5}$ , $\sqrt{7}$ , and $\sqrt[3]{7}$ . These cannot be expressed as exact decimals or simple fractions, making them fundamentally different from rational numbers.

The 6 essential rules for manipulating surds

When working with surds, there are six key rules you need to master. These rules will help you simplify expressions and solve problems involving surds effectively.

Rule 1: Multiplying surds

When you multiply two surds together, you can combine them under a single square root sign.

Rule: $\sqrt{a} \times \sqrt{b} = \sqrt{a \times b}$

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Worked Example: Multiplying Surds

$\sqrt{2} \times \sqrt{3} = \sqrt{2 \times 3} = \sqrt{6}$

This also works when you have the same surd multiplied by itself: $\sqrt{b} \times \sqrt{b} = \sqrt{b \times b} = b$

Rule 2: Dividing surds

When you divide one surd by another, you can write them as a single surd fraction.

Rule: $\sqrt{a} \div \sqrt{b} = \sqrt{a \div b}$

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Worked Example: Dividing Surds

$\sqrt{8} \div \sqrt{2} = \sqrt{8 \div 2} = \sqrt{4} = 2$

Rule 3: Adding and subtracting surds

This is the most important rule to remember: when you're adding or subtracting surds that are different, you simply cannot combine them.

Rule: $\sqrt{a} + \sqrt{b} = \sqrt{a} + \sqrt{b}$ (you cannot simplify this further)

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Critical Point: $\sqrt{a} + \sqrt{b}$ is definitely NOT equal to $\sqrt{a + b}$

This is a common mistake that students make. Always remember that you cannot combine different surds under addition or subtraction.

Rule 4: Squaring brackets containing surds

When you square a bracket that contains a surd, you need to expand it fully.

Rule: $(a + \sqrt{b})^2 = (a + \sqrt{b})(a + \sqrt{b}) = a^2 + 2a\sqrt{b} + b$

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Warning: $(a + \sqrt{b})^2$ is NOT the same as $a^2 + (\sqrt{b})^2$

You must expand the brackets fully using standard algebraic methods to get the correct result.

Rule 5: Multiplying different brackets with surds

When you multiply two different brackets containing surds, you expand them using the standard method.

Rule: $(a + \sqrt{b})(a - \sqrt{b}) = a^2 + a\sqrt{b} - a\sqrt{b} - (\sqrt{b})^2 = a^2 - b$

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Notice how the middle terms cancel out, leaving you with a much simpler expression. This pattern appears frequently in surd problems.

Rule 6: Rationalising the denominator

This technique helps you remove surds from the bottom of fractions, making them easier to work with.

Rule: $\frac{a}{\sqrt{b}} = \frac{a \times \sqrt{b}}{\sqrt{b} \times \sqrt{b}} = \frac{a\sqrt{b}}{b}$

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Key Point: You multiply both the top and bottom of the fraction by the same surd that appears in the denominator. This eliminates the surd from the bottom because $\sqrt{b} \times \sqrt{b} = b$ .

For more complex denominators like $(a \pm \sqrt{b})$ , you multiply by the conjugate $(a \mp \sqrt{b})$ .

Working through examples

Let's look at how these rules work in practice with some detailed examples:

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Worked Example 1: Simplifying Surd Expressions

To write $\sqrt{300} + \sqrt{48} - 2\sqrt{75}$ in the form $a\sqrt{3}$ :

Step 1: Break down each surd into its factors:

$\sqrt{300} = \sqrt{100 \times 3} = 10\sqrt{3}$
$\sqrt{48} = \sqrt{16 \times 3} = 4\sqrt{3}$
$\sqrt{75} = \sqrt{25 \times 3} = 5\sqrt{3}$

Step 2: Substitute and simplify: $\sqrt{300} + \sqrt{48} - 2\sqrt{75} = 10\sqrt{3} + 4\sqrt{3} - 2(5\sqrt{3}) = 10\sqrt{3} + 4\sqrt{3} - 10\sqrt{3} = 4\sqrt{3}$

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Worked Example 2: Finding Exact Values

For a rectangle with length $4x$ cm and width $x$ cm, with area $32$ cm²:

Step 1: Set up the equation: Area = length × width = $4x \times x = 4x^2$

Step 2: Solve for $x$ : Since $4x^2 = 32$ , we get $x^2 = 8$ Therefore $x = \sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$

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Worked Example 3: Rationalising Complex Denominators

To write $\frac{3}{2 + \sqrt{5}}$ in the form $a + b\sqrt{5}$ :

Step 1: Multiply top and bottom by the conjugate $(2 - \sqrt{5})$ : $\frac{3}{2 + \sqrt{5}} \times \frac{2 - \sqrt{5}}{2 - \sqrt{5}} = \frac{3(2 - \sqrt{5})}{(2 + \sqrt{5})(2 - \sqrt{5})}$

Step 2: Simplify the denominator: $(2 + \sqrt{5})(2 - \sqrt{5}) = 4 - 5 = -1$

Step 3: Complete the calculation: $\frac{3(2 - \sqrt{5})}{-1} = \frac{6 - 3\sqrt{5}}{-1} = -6 + 3\sqrt{5}$

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Key Points to Remember:

When multiplying surds, you can combine them under one square root
When dividing surds, you can write them as a single fraction under the square root
You cannot simplify the sum or difference of different surds
When squaring or multiplying brackets with surds, expand them fully using standard algebraic methods
Rationalising the denominator means getting rid of surds from the bottom of fractions by multiplying by the conjugate
Always look for opportunities to simplify surds by factoring out perfect squares

Manipulating Surds (AQA GCSE Maths): Revision Notes