Simplifying and Calculating with Surds

Simplifying Surds

You can easily write down half an apple as 0.5, but try writing down the exact length of the diagonal of a $1 \times 1$ square as a decimal. It goes on forever, which is why we leave it as a square root.
A surd is the square root of a non-square integer. It is an irrational value that must remain in root form to be in its exact form, avoiding decimal rounding.
To reach the simplest form, you must rewrite the surd so the radicand (the number under the root) is as small as possible.
This is achieved by extracting the largest possible square factor using the rule $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ .

Worked Example: Simplifying a Surd

Step 1: Identify the largest square factor of the radicand for $\sqrt{48}$ .

The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The largest square factor is 16.

Step 2: Rewrite the surd as a product.

$\sqrt{48} = \sqrt{16 \times 3}$

Step 3: Separate and calculate the square root of the square factor.

$\sqrt{16} \times \sqrt{3} = 4\sqrt{3}$

Worked Example: Multi-step simplification

If you miss the largest square factor and are simplifying $\sqrt{108}$

Step 1: Use a smaller known square factor, like 9.

$\sqrt{108} = \sqrt{9 \times 12} = 3\sqrt{12}$

Step 2: Simplify the remaining surd further, as 12 has a square factor of 4.

$3\sqrt{4 \times 3} = 3 \times 2\sqrt{3}$

Step 3: Multiply the coefficients for the final answer.

$6\sqrt{3}$

Arithmetic with Surds

Treat surds exactly like algebraic terms when adding or subtracting. Just as $3x + 2x = 5x$ , you can see that $3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2}$ .
You can only add or subtract like surds, which are surds that have the exact same radicand once fully simplified.
The number outside the root is called the coefficient. For multiplication and division, you apply operations to the coefficients and radicands separately.

Worked Example: Addition of Surds

Step 1: Write the expression to calculate exactly.

$\sqrt{12} + \sqrt{27}$

Step 2: Simplify each surd to find a common radicand.

$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$
$\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$

Step 3: Add the coefficients of the like surds.

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

Worked Example: Subtraction of Surds

Step 1: Write the expression.

$\sqrt{54} - \sqrt{24}$

Step 2: Simplify each surd.

$\sqrt{54} = \sqrt{9 \times 6} = 3\sqrt{6}$
$\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$

Step 3: Subtract the coefficients.

$3\sqrt{6} - 2\sqrt{6} = 1\sqrt{6}$ (or simply $\sqrt{6}$ )

Worked Example: Multiplication with Coefficients

Step 1: Write the multiplication expression.

$3\sqrt{2} \times 4\sqrt{10}$

Step 2: Multiply the coefficients and the radicands separately.

$(3 \times 4)\sqrt{2 \times 10} = 12\sqrt{20}$

Step 3: Simplify the resulting surd by finding the largest square factor of 20.

$12\sqrt{4 \times 5} = 12 \times 2\sqrt{5} = 24\sqrt{5}$

Worked Example: Division of Surds

Step 1: Write the division expression.

$\frac{10\sqrt{30}}{2\sqrt{10}}$

Step 2: Divide coefficients and radicands separately.

$(\frac{10}{2})\sqrt{\frac{30}{10}}$

Step 3: Calculate the final exact form.

$5\sqrt{3}$

Expanding Brackets with Surds

When multiplying brackets involving surds, you apply the standard distributive law used in algebra.
It is often helpful to simplify any large surds before expanding to keep the calculations manageable.
A key identity to remember is that squaring a square root removes the root entirely: $(\sqrt{a})^2 = a$ .

Worked Example: Expanding Double Brackets

Step 1: Write the expression to expand.

$(\sqrt{2} + 1)(\sqrt{2} + 3)$

Step 2: Apply the FOIL (First, Outside, Inside, Last) method.

First: $\sqrt{2} \times \sqrt{2} = 2$
Outside: $\sqrt{2} \times 3 = 3\sqrt{2}$
Inside: $1 \times \sqrt{2} = \sqrt{2}$
Last: $1 \times 3 = 3$

Step 3: Collect like terms to simplify.

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

Rationalising the Denominator

Rationalising a fraction means rewriting it so that the denominator is a rational number (an integer), entirely removing any surds from the bottom.
For a simple fraction of the form $\frac{1}{\sqrt{a}}$ , you multiply both the numerator and denominator by $\sqrt{a}$ .
For complex denominators like $a + \sqrt{b}$ , you must multiply by its conjugate pair (the exact same expression, but with the opposite sign). This uses the difference of two squares identity, guaranteeing that the middle surd terms cancel out.

Worked Example: Rationalising a simple denominator

Step 1: Write the fraction.

$\frac{4}{\sqrt{6}}$

Step 2: Multiply the top and bottom by the surd in the denominator.

$\frac{4 \times \sqrt{6}}{\sqrt{6} \times \sqrt{6}}$

Step 3: Calculate the numerator and denominator.

$\frac{4\sqrt{6}}{6}$

Step 4: Simplify the fraction by dividing the coefficient and denominator by their highest common factor (2).

$\frac{2\sqrt{6}}{3}$

Worked Example: Rationalising using a conjugate pair

Step 1: Write the expression.

$\frac{6}{3 + \sqrt{5}}$

Step 2: Multiply the top and bottom by the conjugate pair of the denominator, which is $(3 - \sqrt{5})$ .

$\frac{6(3 - \sqrt{5})}{(3 + \sqrt{5})(3 - \sqrt{5})}$

Step 3: Expand the numerator and denominator. The denominator simplifies using $(a+\sqrt{b})(a-\sqrt{b}) = a^2 - b$ .

Numerator: $18 - 6\sqrt{5}$
Denominator: $9 - 5 = 4$

Step 4: Write the resulting fraction and simplify by dividing all terms by 2.

$\frac{18 - 6\sqrt{5}}{4} = \frac{9 - 3\sqrt{5}}{2}$

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Exam Tips

Common Mistake
Students often incorrectly add radicands together directly (e.g., thinking $\sqrt{9} + \sqrt{4} = \sqrt{13}$ ). You can only add the coefficients of like surds, similar to collecting 'x' terms in algebra.
2
In 'Show that' questions, AQA examiners expect to see the manual extraction steps for simplifying surds; relying entirely on your scientific calculator will lose you method marks.
3
When rationalising with a conjugate pair, always put brackets around the entire numerator to ensure you multiply every single term by the conjugate.
4
If the result of your conjugate pair denominator ( $a^2 - b$ ) evaluates to a negative number, you can move the negative sign up to the numerator to keep the fraction format neat.
5
Always check if your final fraction can be simplified further; leaving an answer as $\frac{10\sqrt{2}}{2}$ instead of $5\sqrt{2}$ will usually cost you the final accuracy mark.

Key Terms(12)

Surd: The square root of a non-square integer, leaving an irrational number that cannot be written as a simple fraction.
Irrational value: A number that cannot be written as a simple fraction, typically having an infinite, non-repeating decimal expansion.
Exact form: A numerical answer expressed using surds or fractions rather than a rounded decimal approximation.
Simplest form: A surd expression where the radicand has no square factors other than 1.
Radicand: The mathematical value or expression contained directly beneath the square root symbol.
Square factor: An integer factor of a number that is itself a perfect square (e.g., 4, 9, 16, 25).
Like surds: Surds that share the exact same radicand once they have been fully simplified.
Coefficient: The rational number placed in front of and multiplying the surd.
Distributive law: The mathematical rule used when expanding brackets, stating that multiplying a number by a group of numbers added together is equivalent to doing each multiplication separately.
Rationalising: The algebraic process of rewriting a fraction so the denominator is an integer, removing any irrational surds from the bottom.
Denominator: The bottom number in a fraction.
Conjugate pair: Two binomial expressions identical except for the opposite arithmetic sign between their terms (e.g., $3 + \sqrt{2}$ and $3 - \sqrt{2}$ ).

Previous NoteExact Calculations with Pi Next Subtopic: Standard FormStandard Form

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Key Terms(12)

Surd: The square root of a non-square integer, leaving an irrational number that cannot be written as a simple fraction.
Irrational value: A number that cannot be written as a simple fraction, typically having an infinite, non-repeating decimal expansion.
Exact form: A numerical answer expressed using surds or fractions rather than a rounded decimal approximation.
Simplest form: A surd expression where the radicand has no square factors other than 1.
Radicand: The mathematical value or expression contained directly beneath the square root symbol.
Square factor: An integer factor of a number that is itself a perfect square (e.g., 4, 9, 16, 25).
Like surds: Surds that share the exact same radicand once they have been fully simplified.
Coefficient: The rational number placed in front of and multiplying the surd.
Distributive law: The mathematical rule used when expanding brackets, stating that multiplying a number by a group of numbers added together is equivalent to doing each multiplication separately.
Rationalising: The algebraic process of rewriting a fraction so the denominator is an integer, removing any irrational surds from the bottom.
Denominator: The bottom number in a fraction.
Conjugate pair: Two binomial expressions identical except for the opposite arithmetic sign between their terms (e.g., $3 + \sqrt{2}$ and $3 - \sqrt{2}$ ).

Simplifying and Calculating with Surds

Simplifying Surds

You can easily write down half an apple as 0.5, but try writing down the exact length of the diagonal of a $1 \times 1$ square as a decimal. It goes on forever, which is why we leave it as a square root.
A surd is the square root of a non-square integer. It is an irrational value that must remain in root form to be in its exact form, avoiding decimal rounding.
To reach the simplest form, you must rewrite the surd so the radicand (the number under the root) is as small as possible.
This is achieved by extracting the largest possible square factor using the rule $\sqrt{ab} = \sqrt{a} \times \sqrt{b}$ .

Worked Example: Simplifying a Surd

Step 1: Identify the largest square factor of the radicand for $\sqrt{48}$ .

The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. The largest square factor is 16.

Step 2: Rewrite the surd as a product.

$\sqrt{48} = \sqrt{16 \times 3}$

Step 3: Separate and calculate the square root of the square factor.

$\sqrt{16} \times \sqrt{3} = 4\sqrt{3}$

Worked Example: Multi-step simplification

If you miss the largest square factor and are simplifying $\sqrt{108}$

Step 1: Use a smaller known square factor, like 9.

$\sqrt{108} = \sqrt{9 \times 12} = 3\sqrt{12}$

Step 2: Simplify the remaining surd further, as 12 has a square factor of 4.

$3\sqrt{4 \times 3} = 3 \times 2\sqrt{3}$

Step 3: Multiply the coefficients for the final answer.

$6\sqrt{3}$

Arithmetic with Surds

Treat surds exactly like algebraic terms when adding or subtracting. Just as $3x + 2x = 5x$ , you can see that $3\sqrt{2} + 2\sqrt{2} = 5\sqrt{2}$ .
You can only add or subtract like surds, which are surds that have the exact same radicand once fully simplified.
The number outside the root is called the coefficient. For multiplication and division, you apply operations to the coefficients and radicands separately.

Worked Example: Addition of Surds

Step 1: Write the expression to calculate exactly.

$\sqrt{12} + \sqrt{27}$

Step 2: Simplify each surd to find a common radicand.

$\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$
$\sqrt{27} = \sqrt{9 \times 3} = 3\sqrt{3}$

Step 3: Add the coefficients of the like surds.

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

Worked Example: Subtraction of Surds

Step 1: Write the expression.

$\sqrt{54} - \sqrt{24}$

Step 2: Simplify each surd.

$\sqrt{54} = \sqrt{9 \times 6} = 3\sqrt{6}$
$\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$

Step 3: Subtract the coefficients.

$3\sqrt{6} - 2\sqrt{6} = 1\sqrt{6}$ (or simply $\sqrt{6}$ )

Worked Example: Multiplication with Coefficients

Step 1: Write the multiplication expression.

$3\sqrt{2} \times 4\sqrt{10}$

Step 2: Multiply the coefficients and the radicands separately.

$(3 \times 4)\sqrt{2 \times 10} = 12\sqrt{20}$

Step 3: Simplify the resulting surd by finding the largest square factor of 20.

$12\sqrt{4 \times 5} = 12 \times 2\sqrt{5} = 24\sqrt{5}$

Worked Example: Division of Surds

Step 1: Write the division expression.

$\frac{10\sqrt{30}}{2\sqrt{10}}$

Step 2: Divide coefficients and radicands separately.

$(\frac{10}{2})\sqrt{\frac{30}{10}}$

Step 3: Calculate the final exact form.

$5\sqrt{3}$

Expanding Brackets with Surds

When multiplying brackets involving surds, you apply the standard distributive law used in algebra.
It is often helpful to simplify any large surds before expanding to keep the calculations manageable.
A key identity to remember is that squaring a square root removes the root entirely: $(\sqrt{a})^2 = a$ .

Worked Example: Expanding Double Brackets

Step 1: Write the expression to expand.

$(\sqrt{2} + 1)(\sqrt{2} + 3)$

Step 2: Apply the FOIL (First, Outside, Inside, Last) method.

First: $\sqrt{2} \times \sqrt{2} = 2$
Outside: $\sqrt{2} \times 3 = 3\sqrt{2}$
Inside: $1 \times \sqrt{2} = \sqrt{2}$
Last: $1 \times 3 = 3$

Step 3: Collect like terms to simplify.

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

Rationalising the Denominator

Rationalising a fraction means rewriting it so that the denominator is a rational number (an integer), entirely removing any surds from the bottom.
For a simple fraction of the form $\frac{1}{\sqrt{a}}$ , you multiply both the numerator and denominator by $\sqrt{a}$ .
For complex denominators like $a + \sqrt{b}$ , you must multiply by its conjugate pair (the exact same expression, but with the opposite sign). This uses the difference of two squares identity, guaranteeing that the middle surd terms cancel out.

Worked Example: Rationalising a simple denominator

Step 1: Write the fraction.

$\frac{4}{\sqrt{6}}$

Step 2: Multiply the top and bottom by the surd in the denominator.

$\frac{4 \times \sqrt{6}}{\sqrt{6} \times \sqrt{6}}$

Step 3: Calculate the numerator and denominator.

$\frac{4\sqrt{6}}{6}$

Step 4: Simplify the fraction by dividing the coefficient and denominator by their highest common factor (2).

$\frac{2\sqrt{6}}{3}$

Worked Example: Rationalising using a conjugate pair

Step 1: Write the expression.

$\frac{6}{3 + \sqrt{5}}$

Step 2: Multiply the top and bottom by the conjugate pair of the denominator, which is $(3 - \sqrt{5})$ .

$\frac{6(3 - \sqrt{5})}{(3 + \sqrt{5})(3 - \sqrt{5})}$

Step 3: Expand the numerator and denominator. The denominator simplifies using $(a+\sqrt{b})(a-\sqrt{b}) = a^2 - b$ .

Numerator: $18 - 6\sqrt{5}$
Denominator: $9 - 5 = 4$

Step 4: Write the resulting fraction and simplify by dividing all terms by 2.

$\frac{18 - 6\sqrt{5}}{4} = \frac{9 - 3\sqrt{5}}{2}$

Sign up to continue reading

Get full access to revision notes, key terms, and exam tips for every subtopic.

Exam Tips

Common Mistake
Students often incorrectly add radicands together directly (e.g., thinking $\sqrt{9} + \sqrt{4} = \sqrt{13}$ ). You can only add the coefficients of like surds, similar to collecting 'x' terms in algebra.
2
In 'Show that' questions, AQA examiners expect to see the manual extraction steps for simplifying surds; relying entirely on your scientific calculator will lose you method marks.
3
When rationalising with a conjugate pair, always put brackets around the entire numerator to ensure you multiply every single term by the conjugate.
4
If the result of your conjugate pair denominator ( $a^2 - b$ ) evaluates to a negative number, you can move the negative sign up to the numerator to keep the fraction format neat.
5
Always check if your final fraction can be simplified further; leaving an answer as $\frac{10\sqrt{2}}{2}$ instead of $5\sqrt{2}$ will usually cost you the final accuracy mark.

Key Terms(12)

Surd: The square root of a non-square integer, leaving an irrational number that cannot be written as a simple fraction.
Irrational value: A number that cannot be written as a simple fraction, typically having an infinite, non-repeating decimal expansion.
Exact form: A numerical answer expressed using surds or fractions rather than a rounded decimal approximation.
Simplest form: A surd expression where the radicand has no square factors other than 1.
Radicand: The mathematical value or expression contained directly beneath the square root symbol.
Square factor: An integer factor of a number that is itself a perfect square (e.g., 4, 9, 16, 25).
Like surds: Surds that share the exact same radicand once they have been fully simplified.
Coefficient: The rational number placed in front of and multiplying the surd.
Distributive law: The mathematical rule used when expanding brackets, stating that multiplying a number by a group of numbers added together is equivalent to doing each multiplication separately.
Rationalising: The algebraic process of rewriting a fraction so the denominator is an integer, removing any irrational surds from the bottom.
Denominator: The bottom number in a fraction.
Conjugate pair: Two binomial expressions identical except for the opposite arithmetic sign between their terms (e.g., $3 + \sqrt{2}$ and $3 - \sqrt{2}$ ).

Previous NoteExact Calculations with Pi Next Subtopic: Standard FormStandard Form

Back to Exact Calculations

Put your knowledge into practice — try past paper questions for Mathematics

Key Terms(12)

Surd: The square root of a non-square integer, leaving an irrational number that cannot be written as a simple fraction.
Irrational value: A number that cannot be written as a simple fraction, typically having an infinite, non-repeating decimal expansion.
Exact form: A numerical answer expressed using surds or fractions rather than a rounded decimal approximation.
Simplest form: A surd expression where the radicand has no square factors other than 1.
Radicand: The mathematical value or expression contained directly beneath the square root symbol.
Square factor: An integer factor of a number that is itself a perfect square (e.g., 4, 9, 16, 25).
Like surds: Surds that share the exact same radicand once they have been fully simplified.
Coefficient: The rational number placed in front of and multiplying the surd.
Distributive law: The mathematical rule used when expanding brackets, stating that multiplying a number by a group of numbers added together is equivalent to doing each multiplication separately.
Rationalising: The algebraic process of rewriting a fraction so the denominator is an integer, removing any irrational surds from the bottom.
Denominator: The bottom number in a fraction.
Conjugate pair: Two binomial expressions identical except for the opposite arithmetic sign between their terms (e.g., $3 + \sqrt{2}$ and $3 - \sqrt{2}$ ).