Surds and Algebraic Fractions

Simplifying Surds

You can write $1 \div 3$ as $0.333...$ and know it repeats a predictable pattern forever, but if you type $\sqrt{3}$ into a calculator, the decimal never ends and never repeats. A surd is the square root of a non-square integer that results in an irrational number. Because decimals lose accuracy, mathematicians leave these as exact values.

To simplify surds, we use the multiplication rule:

\sqrt{ab} = \sqrt{a} \times \sqrt{b}

A surd is in its simplest form when the radicand (the number under the square root sign) has no square factors greater than 1. You must find the largest square number ( $4, 9, 16, 25...$ ) that is a factor of the radicand. You can only add or subtract like surds, which are surds that share the exact same radicand.

Worked Example 1: Basic Simplification

Calculate and simplify $\sqrt{48}$ .

Step 1: Identify the largest square factor of $48$ . ( $48 = 16 \times 3$ )
Step 2: Apply the multiplication rule. ( $\sqrt{48} = \sqrt{16} \times \sqrt{3}$ )
Step 3: Calculate the square root of the square factor. ( $4\sqrt{3}$ )

Worked Example 2: Simplifying with Coefficients

Simplify $3\sqrt{200}$ .

Step 1: Identify the largest square factor of $200$ . ( $200 = 100 \times 2$ )
Step 2: Split the surd, keeping the outside coefficient. ( $3 \times (\sqrt{100} \times \sqrt{2})$ )
Step 3: Calculate and multiply. ( $3 \times 10 \times \sqrt{2} = 30\sqrt{2}$ )

Worked Example 3: Subtraction via Simplification

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

Step 1: Simplify $\sqrt{54}$ . ( $\sqrt{9 \times 6} = 3\sqrt{6}$ )
Step 2: Simplify $\sqrt{24}$ . ( $\sqrt{4 \times 6} = 2\sqrt{6}$ )
Step 3: Subtract the like surds. ( $3\sqrt{6} - 2\sqrt{6} = 1\sqrt{6}$ or $\sqrt{6}$ )

Rationalising the Denominator

Dividing a cake into 3 pieces is easy to picture, but dividing a cake into $\sqrt{3}$ pieces is impossible to visualise. This is why a fraction is never fully simplified if the denominator is a surd. Rationalising the denominator is the process of removing the surd from the bottom to make it a rational number (an integer or simple fraction).

For a single surd, multiply the numerator and denominator by a fraction equal to 1 (e.g., $\frac{\sqrt{3}}{\sqrt{3}}$ ). For a binomial denominator (e.g., $a + \sqrt{b}$ ), you must multiply by its conjugate pair. This is a pair of binomial expressions with the same terms but opposite signs ( $a - \sqrt{b}$ ). This exploits the difference of two squares to ensure the middle surd terms cancel out.

Worked Example 1: Single Term

Rationalise $\frac{6}{\sqrt{2}}$ .

Step 1: Multiply by $\frac{\sqrt{2}}{\sqrt{2}}$ . ( $\frac{6 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{6\sqrt{2}}{2}$ )
Step 2: Simplify the fraction. ( $3\sqrt{2}$ )

Worked Example 2: Binomial Denominator

Rationalise $\frac{7}{2 + \sqrt{3}}$ .

Step 1: Identify the conjugate pair of the denominator. ( $2 - \sqrt{3}$ )
Step 2: Multiply both top and bottom by the conjugate. ( $\frac{7(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})}$ )
Step 3: Expand the denominator using the difference of two squares. ( $2^2 - (\sqrt{3})^2 = 4 - 3 = 1$ )
Step 4: State the final answer. ( $14 - 7\sqrt{3}$ )

Simplifying Algebraic Fractions

Every time you pack a suitcase, you try to condense everything into the smallest possible space. Simplifying an algebraic fraction works exactly the same way: we want the most compact version of an expression.

The golden rule is that you must factorise fully before you cancel. You can only cancel a common factor (brackets or terms that are multiplied). You can NEVER cancel individual terms separated by addition or subtraction. Look out for the difference of two squares in denominators, as AQA frequently tests this.

Worked Example 1: Simplifying with Quadratics

Simplify fully $\frac{2x^2 - 11x + 12}{x^2 - 16}$ .

Step 1: Factorise the numerator into two brackets. ( $(2x - 3)(x - 4)$ )
Step 2: Factorise the denominator using the difference of two squares. ( $(x - 4)(x + 4)$ )
Step 3: Cancel the common factor of $(x - 4)$ . ( $\frac{2x - 3}{x + 4}$ )

Worked Example 2: Simplifying with Single Brackets

Simplify fully $\frac{4x + 6}{2x^2 - 7x - 15}$ .

Step 1: Factorise the linear numerator. ( $2(2x + 3)$ )
Step 2: Factorise the quadratic denominator. ( $(2x + 3)(x - 5)$ )
Step 3: Cancel the common factor of $(2x + 3)$ . ( $\frac{2}{x - 5}$ )

Operating with Algebraic Fractions

How do you add two fractions when the denominators are completely different quadratic equations? Just like numerical fractions, addition and subtraction require finding a common denominator, while multiplication and division have their own distinct rules.

For addition and subtraction, find the lowest common denominator (LCD). If a denominator is a quadratic, factorise it first to find the simplest LCD. For division, multiply the first fraction by the reciprocal (the flipped version) of the second fraction.

Worked Example 1: Standard Addition

Calculate $\frac{4}{x+1} + \frac{3}{x-2}$ .

Step 1: Identify the LCD. ( $(x+1)(x-2)$ )
Step 2: Adjust the numerators by cross-multiplying. ( $\frac{4(x-2) + 3(x+1)}{(x+1)(x-2)}$ )
Step 3: Expand and combine the numerator. ( $\frac{4x - 8 + 3x + 3}{(x+1)(x-2)}$ )
Step 4: Simplify fully. ( $\frac{7x - 5}{(x+1)(x-2)}$ )

Worked Example 2: Subtraction

Calculate $\frac{5}{x-1} - \frac{2}{x+3}$ .

Step 1: Identify the LCD. ( $(x-1)(x+3)$ )
Step 2: Adjust the numerators by cross-multiplying. ( $\frac{5(x+3) - 2(x-1)}{(x-1)(x+3)}$ )
Step 3: Expand the numerator, ensuring the negative sign applies to the entire second bracket. ( $\frac{5x + 15 - 2x + 2}{(x-1)(x+3)}$ )
Step 4: Simplify fully. ( $\frac{3x + 17}{(x-1)(x+3)}$ )

Worked Example 3: Multiplication

Calculate $\frac{x^2 - 9}{2x + 10} \times \frac{x+5}{x-3}$ .

Step 1: Factorise all parts of both fractions. ( $\frac{(x+3)(x-3)}{2(x+5)} \times \frac{x+5}{x-3}$ )
Step 2: Cancel common factors across numerators and denominators. (Cancel $(x-3)$ and $(x+5)$ )
Step 3: State the result. ( $\frac{x+3}{2}$ )

Worked Example 4: Division

Calculate $\frac{x^2 - 4}{3x} \div \frac{x - 2}{6x^2}$ .

Step 1: Rewrite as multiplication by taking the reciprocal of the second fraction. ( $\frac{x^2 - 4}{3x} \times \frac{6x^2}{x - 2}$ )
Step 2: Factorise the terms, including the difference of two squares. ( $\frac{(x-2)(x+2)}{3x} \times \frac{6x^2}{x - 2}$ )
Step 3: Cancel common factors across numerators and denominators. (Cancel $(x-2)$ , and simplify $\frac{6x^2}{3x}$ to $2x$ )
Step 4: Multiply the remaining terms to state the result. ( $2x(x+2)$ )

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Exam Tips

Common Mistake
Students often assume $\sqrt{a} + \sqrt{b} = \sqrt{a+b}$ , or try to cancel individual terms separated by a plus/minus in fractions (like the $x^2$ in $\frac{x^2+5}{x^2}$ ). Only fully factorised, multiplied terms can be cancelled!
2
In AQA 'Show that' questions involving surds or fraction addition, examiners expect you to show every single line of working (like expanding the common denominator) to get the method marks.
3
When rationalising an expression that results in something like $\frac{10 + 2\sqrt{5}}{2}$ , remember you must divide BOTH terms in the numerator by 2 to get the correct final answer of $5 + \sqrt{5}$ .
4
If you are simplifying a fraction and absolutely nothing cancels, stop and re-check your factorisation. Examiners rarely design 3-mark simplification questions with no common factors.
5
When subtracting algebraic fractions, remember the negative sign applies to the ENTIRE numerator of the second fraction, so expand the brackets carefully to avoid sign errors.

Key Terms(12)

Surd: The square root of a non-square integer that results in an irrational number (a decimal that never ends or repeats).
Simplest form: A state where the number under the square root has no square factors greater than 1.
Radicand: The number that is found inside the square root sign.
Like surds: Surds that share the exact same radicand, allowing them to be added or subtracted together.
Rationalising the denominator: The mathematical process of removing a surd from the denominator to make it a rational integer.
Rational number: Any number that can be written as an integer or a simple fraction.
Conjugate pair: A pair of binomial expressions with the exact same terms but opposite signs, used to eliminate surds through expansion.
Algebraic fraction: A fraction that contains variables (letters) in the numerator, the denominator, or both.
Common factor: An expression or bracket that divides exactly into both the entire numerator and the entire denominator.
Difference of two squares: A specific type of quadratic in the form x² - k², which factorises into two brackets: (x-k)(x+k).
Lowest common denominator (LCD): The simplest algebraic expression that is a multiple of all denominators present in an addition or subtraction problem.
Reciprocal: The result of flipping a fraction upside down, turning the numerator into the denominator and vice versa.

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Key Terms(12)

Surd: The square root of a non-square integer that results in an irrational number (a decimal that never ends or repeats).
Simplest form: A state where the number under the square root has no square factors greater than 1.
Radicand: The number that is found inside the square root sign.
Like surds: Surds that share the exact same radicand, allowing them to be added or subtracted together.
Rationalising the denominator: The mathematical process of removing a surd from the denominator to make it a rational integer.
Rational number: Any number that can be written as an integer or a simple fraction.
Conjugate pair: A pair of binomial expressions with the exact same terms but opposite signs, used to eliminate surds through expansion.
Algebraic fraction: A fraction that contains variables (letters) in the numerator, the denominator, or both.
Common factor: An expression or bracket that divides exactly into both the entire numerator and the entire denominator.
Difference of two squares: A specific type of quadratic in the form x² - k², which factorises into two brackets: (x-k)(x+k).
Lowest common denominator (LCD): The simplest algebraic expression that is a multiple of all denominators present in an addition or subtraction problem.
Reciprocal: The result of flipping a fraction upside down, turning the numerator into the denominator and vice versa.

Surds and Algebraic Fractions

Simplifying Surds

To simplify surds, we use the multiplication rule:

\sqrt{ab} = \sqrt{a} \times \sqrt{b}

Worked Example 1: Basic Simplification

Calculate and simplify $\sqrt{48}$ .

Step 1: Identify the largest square factor of $48$ . ( $48 = 16 \times 3$ )
Step 2: Apply the multiplication rule. ( $\sqrt{48} = \sqrt{16} \times \sqrt{3}$ )
Step 3: Calculate the square root of the square factor. ( $4\sqrt{3}$ )

Worked Example 2: Simplifying with Coefficients

Simplify $3\sqrt{200}$ .

Step 1: Identify the largest square factor of $200$ . ( $200 = 100 \times 2$ )
Step 2: Split the surd, keeping the outside coefficient. ( $3 \times (\sqrt{100} \times \sqrt{2})$ )
Step 3: Calculate and multiply. ( $3 \times 10 \times \sqrt{2} = 30\sqrt{2}$ )

Worked Example 3: Subtraction via Simplification

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

Step 1: Simplify $\sqrt{54}$ . ( $\sqrt{9 \times 6} = 3\sqrt{6}$ )
Step 2: Simplify $\sqrt{24}$ . ( $\sqrt{4 \times 6} = 2\sqrt{6}$ )
Step 3: Subtract the like surds. ( $3\sqrt{6} - 2\sqrt{6} = 1\sqrt{6}$ or $\sqrt{6}$ )

Rationalising the Denominator

Worked Example 1: Single Term

Rationalise $\frac{6}{\sqrt{2}}$ .

Step 1: Multiply by $\frac{\sqrt{2}}{\sqrt{2}}$ . ( $\frac{6 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} = \frac{6\sqrt{2}}{2}$ )
Step 2: Simplify the fraction. ( $3\sqrt{2}$ )

Worked Example 2: Binomial Denominator

Rationalise $\frac{7}{2 + \sqrt{3}}$ .

Step 1: Identify the conjugate pair of the denominator. ( $2 - \sqrt{3}$ )
Step 2: Multiply both top and bottom by the conjugate. ( $\frac{7(2 - \sqrt{3})}{(2 + \sqrt{3})(2 - \sqrt{3})}$ )
Step 3: Expand the denominator using the difference of two squares. ( $2^2 - (\sqrt{3})^2 = 4 - 3 = 1$ )
Step 4: State the final answer. ( $14 - 7\sqrt{3}$ )

Simplifying Algebraic Fractions

Worked Example 1: Simplifying with Quadratics

Simplify fully $\frac{2x^2 - 11x + 12}{x^2 - 16}$ .

Step 1: Factorise the numerator into two brackets. ( $(2x - 3)(x - 4)$ )
Step 2: Factorise the denominator using the difference of two squares. ( $(x - 4)(x + 4)$ )
Step 3: Cancel the common factor of $(x - 4)$ . ( $\frac{2x - 3}{x + 4}$ )

Worked Example 2: Simplifying with Single Brackets

Simplify fully $\frac{4x + 6}{2x^2 - 7x - 15}$ .

Step 1: Factorise the linear numerator. ( $2(2x + 3)$ )
Step 2: Factorise the quadratic denominator. ( $(2x + 3)(x - 5)$ )
Step 3: Cancel the common factor of $(2x + 3)$ . ( $\frac{2}{x - 5}$ )

Operating with Algebraic Fractions

Worked Example 1: Standard Addition

Calculate $\frac{4}{x+1} + \frac{3}{x-2}$ .

Step 1: Identify the LCD. ( $(x+1)(x-2)$ )
Step 2: Adjust the numerators by cross-multiplying. ( $\frac{4(x-2) + 3(x+1)}{(x+1)(x-2)}$ )
Step 3: Expand and combine the numerator. ( $\frac{4x - 8 + 3x + 3}{(x+1)(x-2)}$ )
Step 4: Simplify fully. ( $\frac{7x - 5}{(x+1)(x-2)}$ )

Worked Example 2: Subtraction

Calculate $\frac{5}{x-1} - \frac{2}{x+3}$ .

Step 1: Identify the LCD. ( $(x-1)(x+3)$ )
Step 2: Adjust the numerators by cross-multiplying. ( $\frac{5(x+3) - 2(x-1)}{(x-1)(x+3)}$ )
Step 3: Expand the numerator, ensuring the negative sign applies to the entire second bracket. ( $\frac{5x + 15 - 2x + 2}{(x-1)(x+3)}$ )
Step 4: Simplify fully. ( $\frac{3x + 17}{(x-1)(x+3)}$ )

Worked Example 3: Multiplication

Calculate $\frac{x^2 - 9}{2x + 10} \times \frac{x+5}{x-3}$ .

Step 1: Factorise all parts of both fractions. ( $\frac{(x+3)(x-3)}{2(x+5)} \times \frac{x+5}{x-3}$ )
Step 2: Cancel common factors across numerators and denominators. (Cancel $(x-3)$ and $(x+5)$ )
Step 3: State the result. ( $\frac{x+3}{2}$ )

Worked Example 4: Division

Calculate $\frac{x^2 - 4}{3x} \div \frac{x - 2}{6x^2}$ .

Step 1: Rewrite as multiplication by taking the reciprocal of the second fraction. ( $\frac{x^2 - 4}{3x} \times \frac{6x^2}{x - 2}$ )
Step 2: Factorise the terms, including the difference of two squares. ( $\frac{(x-2)(x+2)}{3x} \times \frac{6x^2}{x - 2}$ )
Step 3: Cancel common factors across numerators and denominators. (Cancel $(x-2)$ , and simplify $\frac{6x^2}{3x}$ to $2x$ )
Step 4: Multiply the remaining terms to state the result. ( $2x(x+2)$ )

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 assume $\sqrt{a} + \sqrt{b} = \sqrt{a+b}$ , or try to cancel individual terms separated by a plus/minus in fractions (like the $x^2$ in $\frac{x^2+5}{x^2}$ ). Only fully factorised, multiplied terms can be cancelled!
2
In AQA 'Show that' questions involving surds or fraction addition, examiners expect you to show every single line of working (like expanding the common denominator) to get the method marks.
3
When rationalising an expression that results in something like $\frac{10 + 2\sqrt{5}}{2}$ , remember you must divide BOTH terms in the numerator by 2 to get the correct final answer of $5 + \sqrt{5}$ .
4
If you are simplifying a fraction and absolutely nothing cancels, stop and re-check your factorisation. Examiners rarely design 3-mark simplification questions with no common factors.
5
When subtracting algebraic fractions, remember the negative sign applies to the ENTIRE numerator of the second fraction, so expand the brackets carefully to avoid sign errors.

Key Terms(12)

Surd: The square root of a non-square integer that results in an irrational number (a decimal that never ends or repeats).
Simplest form: A state where the number under the square root has no square factors greater than 1.
Radicand: The number that is found inside the square root sign.
Like surds: Surds that share the exact same radicand, allowing them to be added or subtracted together.
Rationalising the denominator: The mathematical process of removing a surd from the denominator to make it a rational integer.
Rational number: Any number that can be written as an integer or a simple fraction.
Conjugate pair: A pair of binomial expressions with the exact same terms but opposite signs, used to eliminate surds through expansion.
Algebraic fraction: A fraction that contains variables (letters) in the numerator, the denominator, or both.
Common factor: An expression or bracket that divides exactly into both the entire numerator and the entire denominator.
Difference of two squares: A specific type of quadratic in the form x² - k², which factorises into two brackets: (x-k)(x+k).
Lowest common denominator (LCD): The simplest algebraic expression that is a multiple of all denominators present in an addition or subtraction problem.
Reciprocal: The result of flipping a fraction upside down, turning the numerator into the denominator and vice versa.

Previous NoteExpanding and Factorising Quadratics Next Subtopic: Standard FormulaeStandard Formulae

Back to Expression Manipulation

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

Key Terms(12)

Surd: The square root of a non-square integer that results in an irrational number (a decimal that never ends or repeats).
Simplest form: A state where the number under the square root has no square factors greater than 1.
Radicand: The number that is found inside the square root sign.
Like surds: Surds that share the exact same radicand, allowing them to be added or subtracted together.
Rationalising the denominator: The mathematical process of removing a surd from the denominator to make it a rational integer.
Rational number: Any number that can be written as an integer or a simple fraction.
Conjugate pair: A pair of binomial expressions with the exact same terms but opposite signs, used to eliminate surds through expansion.
Algebraic fraction: A fraction that contains variables (letters) in the numerator, the denominator, or both.
Common factor: An expression or bracket that divides exactly into both the entire numerator and the entire denominator.
Difference of two squares: A specific type of quadratic in the form x² - k², which factorises into two brackets: (x-k)(x+k).
Lowest common denominator (LCD): The simplest algebraic expression that is a multiple of all denominators present in an addition or subtraction problem.
Reciprocal: The result of flipping a fraction upside down, turning the numerator into the denominator and vice versa.