Surds

What you'll learn

Recognise a surd and simplify square roots into exact form.
Expand brackets containing surds.
Rationalise denominators so there are no roots on the bottom.
Use the same ideas with simple algebraic surds.

Square roots and surds

A square number is made by multiplying an integer by itself. For example, 36 is a square number because 6 times 6 is 36.

An integer is a whole number: it can be positive, negative or zero.

A rational number can be written as a fraction of integers. An irrational number cannot be written exactly as a fraction.

Definition

Surd

A surd is an irrational root left in exact form, such as $2\sqrt{2}$ or $5\sqrt{5}$ . A root like $36\sqrt{36}$ is not a surd because it simplifies to 6.

Simplifying surds

To simplify a surd, look for the largest square number that is a factor.

Key Idea

Simplest surd form

Factor out the largest square number first. This gets you to the simplest form quickly and avoids doing extra work.

Example

Writing $72$ in the form $k 2$

Factorising 72 into its largest square factor shows why the root simplifies cleanly.

Find the largest square factor of 72. Since 72 is 36 times 2, write:
$72=36×2\sqrt{72}=\sqrt{36 \times 2}$
Split the square root into two parts:
$36×2=362\sqrt{36 \times 2}=\sqrt{36}\sqrt{2}$
Simplify $36\sqrt{36}$ :
$72=62\sqrt{72}=6\sqrt{2}$

Common Mistake

Splitting addition

You may split multiplication inside a square root, but not addition. For example, $9+16≠9+16\sqrt{9+16}\neq \sqrt{9}+\sqrt{16}$ .

Surds with coefficients

A coefficient is the number multiplying an expression. In $535\sqrt{3}$ , the coefficient is 5.

When there is already a number in front of the surd, simplify the root first, then multiply the coefficients.

Example

Writing $545$ in the form $k 5$

The coefficient stays outside while the root part is simplified first.

Simplify the square root part:
$45=9×5=35\sqrt{45}=\sqrt{9 \times 5}=3\sqrt{5}$
Put this back into the expression:
$545=5×355\sqrt{45}=5 \times 3\sqrt{5}$
Multiply the coefficients:
$545=1555\sqrt{45}=15\sqrt{5}$

Expanding brackets with surds

To expand means to multiply out the brackets. Treat surds like algebra terms, but remember that a surd times itself becomes a whole number, for example $7×7=7\sqrt{7}\times\sqrt{7}=7$ .

Like surds have the same root part, such as $323\sqrt{2}$ and $−52-5\sqrt{2}$ . You can add or subtract like surds.

Example

Expanding $(3 + 2) (2 - 2)$

A grid helps make sure all four products are included when expanding brackets with surds.

Multiply each term in the first bracket by each term in the second bracket:
$(3+2)(2−2)(3+\sqrt{2})(2-\sqrt{2})$
Write out the four products:
$6−32+22−26-3\sqrt{2}+2\sqrt{2}-2$
Combine the number parts and the surd parts:
$4−24-\sqrt{2}$

Squaring a bracket

Squaring a bracket means multiplying the bracket by itself. Do not just square the two terms separately — there is usually a middle term.

Example

Writing $(5 - 3)^{2}$ in the form $a + b 3$

Writing the square as two brackets makes the two middle terms visible.

Rewrite the square as two brackets:
$(5−3)2=(5−3)(5−3)(5-\sqrt{3})^2=(5-\sqrt{3})(5-\sqrt{3})$
Expand carefully:
$25−53−53+325-5\sqrt{3}-5\sqrt{3}+3$
Collect like terms:
$28−10328-10\sqrt{3}$

Common Mistake

Forgetting the middle terms

The expression $(5−3)2(5-\sqrt{3})^2$ is not $25 + 3$ . The two middle terms, $−53-5\sqrt{3}$ and $−53-5\sqrt{3}$ , must be included.

Conjugates

Definition

Conjugates

Conjugates are two expressions that differ only by the sign between the terms, such as $4+74+\sqrt{7}$ and $4−74-\sqrt{7}$ .

Conjugates are useful because the surd parts cancel when multiplied:

Multiplying conjugates cancels the surd terms and leaves a difference of two squares.

(a+b)(a−b)=a2−b(a+\sqrt{b})(a-\sqrt{b})=a^2-b

Example

Expanding $(7 + 23) (7 - 23)$

Notice that the brackets are conjugates, so use difference of two squares:
$(7+23)(7−23)=72−(23)2(7+2\sqrt{3})(7-2\sqrt{3})=7^2-(2\sqrt{3})^2$
Square each part:
$\times 3)$
Simplify:
$37$

Rationalising denominators

Definition

Rationalising the denominator

To rationalise the denominator means to rewrite a fraction so that there is no surd on the bottom.

If the denominator is a single surd, multiply the top and bottom by that surd.

Example

Simplifying $\frac{4 + 8}{2}$

Multiplying by the same surd over itself removes the surd from a single-term denominator.

Multiply the numerator and denominator by $2\sqrt{2}$ :
$4+82×22\frac{4+\sqrt{8}}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}$
Expand the numerator and simplify the denominator:
$42+162\frac{4\sqrt{2}+\sqrt{16}}{2}$
Simplify fully:
$42+42=22+2\frac{4\sqrt{2}+4}{2}=2\sqrt{2}+2$

Rationalising with a conjugate

If the denominator has two terms, such as $2+32+\sqrt{3}$ , multiply by its conjugate, $2−32-\sqrt{3}$ .

Example

Showing $\frac{4 + 3}{2 + 3} = 5 - 23$

For a two-term denominator, multiplying by the conjugate makes the surd parts cancel.

Multiply the top and bottom by the conjugate of the denominator:
$4+32+3×2−32−3\frac{4+\sqrt{3}}{2+\sqrt{3}}\times\frac{2-\sqrt{3}}{2-\sqrt{3}}$
Expand the denominator:
$(2+3)(2−3)=4−3=1(2+\sqrt{3})(2-\sqrt{3})=4-3=1$
Expand the numerator:
$(4+3)(2−3)=8−43+23−3(4+\sqrt{3})(2-\sqrt{3})=8-4\sqrt{3}+2\sqrt{3}-3$
Simplify:
$5−235-2\sqrt{3}$

Tip

Choosing what to multiply by

For a two-term denominator, change only the sign in the middle. The conjugate of $3−53-\sqrt{5}$ is $3+53+\sqrt{5}$ .

Fractions inside fractions

Sometimes the denominator contains a small fraction. First combine the denominator into one fraction, then simplify.

Example

Simplifying $\frac{1}{\frac{1}{5} + 5}$

$Combining the denominator into one fraction first makes the complex fraction easier to simplify.$

Write both parts of the denominator over $5\sqrt{5}$ :
$15+5=15+55\frac{1}{\sqrt{5}}+\sqrt{5}=\frac{1}{\sqrt{5}}+\frac{5}{\sqrt{5}}$
Add the fractions in the denominator:
$15+5=65\frac{1}{\sqrt{5}}+\sqrt{5}=\frac{6}{\sqrt{5}}$
Divide by a fraction by multiplying by its reciprocal:
$165=56\frac{1}{\frac{6}{\sqrt{5}}}=\frac{\sqrt{5}}{6}$

Algebraic surds

The same rules work with letters. For GCSE questions, assume the expressions under square roots are non-negative unless told otherwise.

Common Mistake

Variables under roots

A square root such as $x\sqrt{x}$ only makes sense in GCSE real-number work when $x≥0x\ge 0$ . Also, denominators must not be zero.

Example

Simplifying algebraic surds

Use conjugates to simplify $(m+n)(m−n)(\sqrt{m}+\sqrt{n})(\sqrt{m}-\sqrt{n})$ :
$(m+n)(m−n)=m−n(\sqrt{m}+\sqrt{n})(\sqrt{m}-\sqrt{n})=m-n$
Expand $(3p+q)2(3p+\sqrt{q})^2$ by writing it as two brackets:
$(3p+q)2=(3p+q)(3p+q)(3p+\sqrt{q})^2=(3p+\sqrt{q})(3p+\sqrt{q})$
Multiply out and collect terms:
$9p2+6pq+q9p^2+6p\sqrt{q}+q$

Exam technique

In the exam

Look for square factors first when simplifying a single surd.
When expanding brackets, write all four products before collecting terms.
To rationalise a two-term denominator, multiply by the conjugate and simplify carefully.

Self review

Check yourself

Can you simplify a surd by finding the largest square factor?
Can you expand a squared bracket without losing the middle terms?
Can you choose the correct conjugate to rationalise a denominator?

Surds

What you'll learn

Recognise a surd and simplify square roots into exact form.
Expand brackets containing surds.
Rationalise denominators so there are no roots on the bottom.
Use the same ideas with simple algebraic surds.

Square roots and surds

A square number is made by multiplying an integer by itself. For example, 36 is a square number because 6 times 6 is 36.

An integer is a whole number: it can be positive, negative or zero.

A rational number can be written as a fraction of integers. An irrational number cannot be written exactly as a fraction.

Definition

Surd

A surd is an irrational root left in exact form, such as $2\sqrt{2}$ or $5\sqrt{5}$ . A root like $36\sqrt{36}$ is not a surd because it simplifies to 6.

Simplifying surds

To simplify a surd, look for the largest square number that is a factor.

Key Idea

Simplest surd form

Factor out the largest square number first. This gets you to the simplest form quickly and avoids doing extra work.

Example

Writing $72$ in the form $k 2$

Factorising 72 into its largest square factor shows why the root simplifies cleanly.

Find the largest square factor of 72. Since 72 is 36 times 2, write:
$72=36×2\sqrt{72}=\sqrt{36 \times 2}$
Split the square root into two parts:
$36×2=362\sqrt{36 \times 2}=\sqrt{36}\sqrt{2}$
Simplify $36\sqrt{36}$ :
$72=62\sqrt{72}=6\sqrt{2}$

Common Mistake

Splitting addition

You may split multiplication inside a square root, but not addition. For example, $9+16≠9+16\sqrt{9+16}\neq \sqrt{9}+\sqrt{16}$ .

Surds with coefficients

A coefficient is the number multiplying an expression. In $535\sqrt{3}$ , the coefficient is 5.

When there is already a number in front of the surd, simplify the root first, then multiply the coefficients.

Example

Writing $545$ in the form $k 5$

The coefficient stays outside while the root part is simplified first.

Simplify the square root part:
$45=9×5=35\sqrt{45}=\sqrt{9 \times 5}=3\sqrt{5}$
Put this back into the expression:
$545=5×355\sqrt{45}=5 \times 3\sqrt{5}$
Multiply the coefficients:
$545=1555\sqrt{45}=15\sqrt{5}$

Expanding brackets with surds

To expand means to multiply out the brackets. Treat surds like algebra terms, but remember that a surd times itself becomes a whole number, for example $7×7=7\sqrt{7}\times\sqrt{7}=7$ .

Like surds have the same root part, such as $323\sqrt{2}$ and $−52-5\sqrt{2}$ . You can add or subtract like surds.

Example

Expanding $(3 + 2) (2 - 2)$

A grid helps make sure all four products are included when expanding brackets with surds.

Multiply each term in the first bracket by each term in the second bracket:
$(3+2)(2−2)(3+\sqrt{2})(2-\sqrt{2})$
Write out the four products:
$6−32+22−26-3\sqrt{2}+2\sqrt{2}-2$
Combine the number parts and the surd parts:
$4−24-\sqrt{2}$

Squaring a bracket

Squaring a bracket means multiplying the bracket by itself. Do not just square the two terms separately — there is usually a middle term.

Example

Writing $(5 - 3)^{2}$ in the form $a + b 3$

Writing the square as two brackets makes the two middle terms visible.

Rewrite the square as two brackets:
$(5−3)2=(5−3)(5−3)(5-\sqrt{3})^2=(5-\sqrt{3})(5-\sqrt{3})$
Expand carefully:
$25−53−53+325-5\sqrt{3}-5\sqrt{3}+3$
Collect like terms:
$28−10328-10\sqrt{3}$

Common Mistake

Forgetting the middle terms

The expression $(5−3)2(5-\sqrt{3})^2$ is not $25 + 3$ . The two middle terms, $−53-5\sqrt{3}$ and $−53-5\sqrt{3}$ , must be included.

Conjugates

Definition

Conjugates

Conjugates are two expressions that differ only by the sign between the terms, such as $4+74+\sqrt{7}$ and $4−74-\sqrt{7}$ .

Conjugates are useful because the surd parts cancel when multiplied:

Multiplying conjugates cancels the surd terms and leaves a difference of two squares.

(a+b)(a−b)=a2−b(a+\sqrt{b})(a-\sqrt{b})=a^2-b

Example

Expanding $(7 + 23) (7 - 23)$

Notice that the brackets are conjugates, so use difference of two squares:
$(7+23)(7−23)=72−(23)2(7+2\sqrt{3})(7-2\sqrt{3})=7^2-(2\sqrt{3})^2$
Square each part:
$\times 3)$
Simplify:
$37$

Rationalising denominators

Definition

Rationalising the denominator

To rationalise the denominator means to rewrite a fraction so that there is no surd on the bottom.

If the denominator is a single surd, multiply the top and bottom by that surd.

Example

Simplifying $\frac{4 + 8}{2}$

Multiplying by the same surd over itself removes the surd from a single-term denominator.

Multiply the numerator and denominator by $2\sqrt{2}$ :
$4+82×22\frac{4+\sqrt{8}}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}$
Expand the numerator and simplify the denominator:
$42+162\frac{4\sqrt{2}+\sqrt{16}}{2}$
Simplify fully:
$42+42=22+2\frac{4\sqrt{2}+4}{2}=2\sqrt{2}+2$

Rationalising with a conjugate

If the denominator has two terms, such as $2+32+\sqrt{3}$ , multiply by its conjugate, $2−32-\sqrt{3}$ .

Example

Showing $\frac{4 + 3}{2 + 3} = 5 - 23$

For a two-term denominator, multiplying by the conjugate makes the surd parts cancel.

Multiply the top and bottom by the conjugate of the denominator:
$4+32+3×2−32−3\frac{4+\sqrt{3}}{2+\sqrt{3}}\times\frac{2-\sqrt{3}}{2-\sqrt{3}}$
Expand the denominator:
$(2+3)(2−3)=4−3=1(2+\sqrt{3})(2-\sqrt{3})=4-3=1$
Expand the numerator:
$(4+3)(2−3)=8−43+23−3(4+\sqrt{3})(2-\sqrt{3})=8-4\sqrt{3}+2\sqrt{3}-3$
Simplify:
$5−235-2\sqrt{3}$

Tip

Choosing what to multiply by

For a two-term denominator, change only the sign in the middle. The conjugate of $3−53-\sqrt{5}$ is $3+53+\sqrt{5}$ .

Fractions inside fractions

Sometimes the denominator contains a small fraction. First combine the denominator into one fraction, then simplify.

Example

Simplifying $\frac{1}{\frac{1}{5} + 5}$

$Combining the denominator into one fraction first makes the complex fraction easier to simplify.$

Write both parts of the denominator over $5\sqrt{5}$ :
$15+5=15+55\frac{1}{\sqrt{5}}+\sqrt{5}=\frac{1}{\sqrt{5}}+\frac{5}{\sqrt{5}}$
Add the fractions in the denominator:
$15+5=65\frac{1}{\sqrt{5}}+\sqrt{5}=\frac{6}{\sqrt{5}}$
Divide by a fraction by multiplying by its reciprocal:
$165=56\frac{1}{\frac{6}{\sqrt{5}}}=\frac{\sqrt{5}}{6}$

Algebraic surds

The same rules work with letters. For GCSE questions, assume the expressions under square roots are non-negative unless told otherwise.

Common Mistake

Variables under roots

A square root such as $x\sqrt{x}$ only makes sense in GCSE real-number work when $x≥0x\ge 0$ . Also, denominators must not be zero.

Example

Simplifying algebraic surds

Use conjugates to simplify $(m+n)(m−n)(\sqrt{m}+\sqrt{n})(\sqrt{m}-\sqrt{n})$ :
$(m+n)(m−n)=m−n(\sqrt{m}+\sqrt{n})(\sqrt{m}-\sqrt{n})=m-n$
Expand $(3p+q)2(3p+\sqrt{q})^2$ by writing it as two brackets:
$(3p+q)2=(3p+q)(3p+q)(3p+\sqrt{q})^2=(3p+\sqrt{q})(3p+\sqrt{q})$
Multiply out and collect terms:
$9p2+6pq+q9p^2+6p\sqrt{q}+q$

Exam technique

In the exam

Look for square factors first when simplifying a single surd.
When expanding brackets, write all four products before collecting terms.
To rationalise a two-term denominator, multiply by the conjugate and simplify carefully.

Self review

Check yourself

Can you simplify a surd by finding the largest square factor?
Can you expand a squared bracket without losing the middle terms?
Can you choose the correct conjugate to rationalise a denominator?

Surds

What you'll learn

Square roots and surds

Simplifying surds

Surds with coefficients

Expanding brackets with surds

Squaring a bracket

Conjugates

Rationalising denominators

Rationalising with a conjugate

Fractions inside fractions

Algebraic surds

Recap questions

You've reached the end

Surds

What you'll learn

Square roots and surds

Simplifying surds

Surds with coefficients

Expanding brackets with surds

Squaring a bracket

Conjugates

Rationalising denominators

Rationalising with a conjugate

Fractions inside fractions

Algebraic surds

Recap questions

You've reached the end

Surds

What you'll learn

Square roots and surds

Simplifying surds

Surds with coefficients

Expanding brackets with surds

Squaring a bracket

Conjugates

Rationalising denominators

Rationalising with a conjugate

Fractions inside fractions

Algebraic surds

Recap questions

You've reached the end

Surds Revision Guide

Surds

What you'll learn

Square roots and surds

Simplifying surds

Surds with coefficients

Expanding brackets with surds

Squaring a bracket

Conjugates

Rationalising denominators

Rationalising with a conjugate

Fractions inside fractions

Algebraic surds

Recap questions

You've reached the end

Surds Revision Guide