Surds

What you'll learn

Recognise what a surd is and simplify square roots exactly.
Expand brackets involving surds and collect like terms.
Rationalise denominators, including when the denominator has two terms.
Use the same rules with algebraic surds such as $x\sqrt{x}$ .

1. What is a surd?

Before surds, you need to be confident with square numbers: 1, 4, 9, 16, 25, 36, 49, and so on.

A square root asks: “What number squares to give this?” For example, $36=6\sqrt{36}=6$ because $6^2=36$ .

Definition

Key vocabulary

A perfect square is a number made by squaring an integer, such as 16 or 81.
An irrational number cannot be written exactly as a fraction of integers.
A surd is an exact root that is irrational, such as $2\sqrt{2}$ , $7\sqrt{7}$ , or $45\sqrt{45}$ .

So $25\sqrt{25}$ is not a surd because it equals 5, but $5\sqrt{5}$ is a surd.

Example

Recognising surds

Decide which of $64\sqrt{64}$ , $18\sqrt{18}$ , and $10\sqrt{10}$ are surds.

First check for exact square roots:
$64=8\sqrt{64}=8$
Since $64\sqrt{64}$ is a whole number, it is not a surd.
The numbers 18 and 10 are not perfect squares, so $18\sqrt{18}$ and $10\sqrt{10}$ are surds.

2. Simplifying surds

To simplify a surd, look for the largest perfect-square factor inside the square root.

Key Idea

Pull out square factors

For non-negative values, you can split a product inside a square root: $ab=ab\sqrt{ab}=\sqrt{a}\sqrt{b}$ . This lets you take square factors out of the root.

For example, if a number contains a factor of 4, 9, 16, 25, 36, and so on, that factor can usually help you simplify.

Example

Writing a surd in simplest form

Write $72\sqrt{72}$ in the form $k2k\sqrt{2}$ , where $k$ is an integer.

Find a square factor of 72 that leaves 2:
$\times 2$
Split the square root and simplify the square part:
$72=36×2=362=62\sqrt{72}=\sqrt{36 \times 2}=\sqrt{36}\sqrt{2}=6\sqrt{2}$
Therefore the answer is $626\sqrt{2}$ .

Surds with a number in front

If there is already a number multiplying the surd, simplify the root first, then multiply the outside numbers.

Example

Simplifying with a coefficient

Write $4454\sqrt{45}$ in the form $k5k\sqrt{5}$ , where $k$ is an integer.

Simplify the surd part first:
$45=9×5=35\sqrt{45}=\sqrt{9 \times 5}=3\sqrt{5}$
Now multiply by the coefficient 4:
$445=4×35=1254\sqrt{45}=4 \times 3\sqrt{5}=12\sqrt{5}$
So $k = 12$ .

Common Mistake

Do not split sums inside roots

You may split multiplication, but not addition. For example, $9+16\sqrt{9+16}$ is not the same as $9+16\sqrt{9}+\sqrt{16}$ .

Like surds

Definition

Like surds

Like surds have the same root part, such as $373\sqrt{7}$ and $575\sqrt{7}$ . You can add or subtract like surds, just like collecting like terms in algebra.

For example, $23+73=932\sqrt{3}+7\sqrt{3}=9\sqrt{3}$ , but $23+752\sqrt{3}+7\sqrt{5}$ cannot be collected.

Example

Collecting like surds

Simplify $348+2753\sqrt{48}+2\sqrt{75}$ .

Simplify each surd separately:
$48=16×3=43\sqrt{48}=\sqrt{16 \times 3}=4\sqrt{3}$
Also simplify $75\sqrt{75}$ :
$75=25×3=53\sqrt{75}=\sqrt{25 \times 3}=5\sqrt{3}$
Substitute these back in:
$348+275=3(43)+2(53)3\sqrt{48}+2\sqrt{75}=3(4\sqrt{3})+2(5\sqrt{3})$
Collect the like surds:
$123+103=22312\sqrt{3}+10\sqrt{3}=22\sqrt{3}$

3. Expanding brackets with surds

Expand brackets with surds in the same way as algebra brackets: multiply every term in the first bracket by every term in the second bracket.

Remember that $aa=a\sqrt{a}\sqrt{a}=a$ . For example, $55=5\sqrt{5}\sqrt{5}=5$ .

Example

Expanding two brackets

Expand and simplify $(2+7)(3−7)(2+\sqrt{7})(3-\sqrt{7})$ .

Multiply out the four products:
$(2+7)(3−7)=6−27+37−(7)2(2+\sqrt{7})(3-\sqrt{7})=6-2\sqrt{7}+3\sqrt{7}-(\sqrt{7})^2$
Simplify the square root squared:
$6−27+37−76-2\sqrt{7}+3\sqrt{7}-7$
Collect the ordinary numbers and the like surds:
$−1+7-1+\sqrt{7}$

Conjugates

Definition

Conjugates

Two expressions such as $a+ba+\sqrt{b}$ and $a−ba-\sqrt{b}$ are called conjugates. Their product has no surd part because the middle terms cancel.

This is just the difference of two squares:

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

Example

Using conjugates

Expand and simplify $(5+6)(5−6)(5+\sqrt{6})(5-\sqrt{6})$ .

Recognise the brackets as conjugates.
Use the difference of two squares:
$(5+6)(5−6)=52−(6)2(5+\sqrt{6})(5-\sqrt{6})=5^2-(\sqrt{6})^2$
Simplify:
$25 - 6 = 19$

Squaring a bracket

When you square a bracket, write it twice. This helps you avoid missing the middle term.

Example

Squaring a surd bracket

Write $(3−5)2(3-\sqrt{5})^2$ in the form $a+b5a+b\sqrt{5}$ .

Write the squared bracket as two identical brackets:
$(3−5)2=(3−5)(3−5)(3-\sqrt{5})^2=(3-\sqrt{5})(3-\sqrt{5})$
Expand:
$9−35−35+59-3\sqrt{5}-3\sqrt{5}+5$
Collect terms:
$14−6514-6\sqrt{5}$

Common Mistake

Missing the middle term

$(3−5)2(3-\sqrt{5})^2$ is not just $32+(5)23^2+(\sqrt{5})^2$ . The two middle terms are essential.

4. Rationalising the denominator

Definition

Rationalising the denominator

To rationalise the denominator means rewriting a fraction so there is no surd in the denominator.

A fraction is not considered fully simplified if the denominator still contains a surd.

Single surd denominator

If the denominator is just one square root, multiply the numerator and denominator by that square root.

Example

Rationalising a single surd denominator

Rationalise $123\frac{12}{\sqrt{3}}$ .

Multiply top and bottom by $3\sqrt{3}$ :
$123×33=1233\frac{12}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=\frac{12\sqrt{3}}{3}$
Simplify the fraction:
$1233=43\frac{12\sqrt{3}}{3}=4\sqrt{3}$

If the numerator has more than one term, multiply the whole numerator by the surd.

Example

Rationalising with a bracket on top

Simplify fully $4+62\frac{4+\sqrt{6}}{\sqrt{2}}$ .

Multiply the numerator and denominator by $2\sqrt{2}$ :
$4+62×22=42+122\frac{4+\sqrt{6}}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{4\sqrt{2}+\sqrt{12}}{2}$
Simplify $12\sqrt{12}$ :
$12=23\sqrt{12}=2\sqrt{3}$
Divide both terms in the numerator by 2:
$42+232=22+3\frac{4\sqrt{2}+2\sqrt{3}}{2}=2\sqrt{2}+\sqrt{3}$

Two-term denominator

If the denominator is something like $3+23+\sqrt{2}$ , multiply by its conjugate $3−23-\sqrt{2}$ .

Tip

Choosing the multiplier

For a one-term denominator, multiply by the same surd. For a two-term denominator, multiply by the conjugate.

Example

Rationalising a two-term denominator

Show that $10+23+2\frac{10+\sqrt{2}}{3+\sqrt{2}}$ simplifies to $4−24-\sqrt{2}$ .

Multiply top and bottom by the conjugate of the denominator:
$10+23+2×3−23−2\frac{10+\sqrt{2}}{3+\sqrt{2}}\times\frac{3-\sqrt{2}}{3-\sqrt{2}}$
The denominator becomes a difference of two squares:
$(3+2)(3−2)=9−2=7(3+\sqrt{2})(3-\sqrt{2})=9-2=7$
Expand the numerator:
$(10+2)(3−2)=30−102+32−2=28−72(10+\sqrt{2})(3-\sqrt{2})=30-10\sqrt{2}+3\sqrt{2}-2=28-7\sqrt{2}$
Divide by 7:
$28−727=4−2\frac{28-7\sqrt{2}}{7}=4-\sqrt{2}$

Common Mistake

Only changing the denominator

You must multiply the numerator and denominator by the same expression. Otherwise you change the value of the fraction.

Fractions inside fractions

Sometimes the denominator contains a fraction, such as $12+1\frac{1}{\sqrt{2}}+1$ . First combine the denominator, then rationalise if needed.

Example

Simplifying a fraction inside a fraction

Simplify $312+1\frac{3}{\frac{1}{\sqrt{2}}+1}$ .

Write the denominator as one fraction:
$12+1=1+22\frac{1}{\sqrt{2}}+1=\frac{1+\sqrt{2}}{\sqrt{2}}$
Divide by this fraction by multiplying by its reciprocal:
$31+22=321+2\frac{3}{\frac{1+\sqrt{2}}{\sqrt{2}}}=\frac{3\sqrt{2}}{1+\sqrt{2}}$
Rationalise using the conjugate $1−21-\sqrt{2}$ :
$321+2×1−21−2=32−6−1=6−32\frac{3\sqrt{2}}{1+\sqrt{2}}\times\frac{1-\sqrt{2}}{1-\sqrt{2}}=\frac{3\sqrt{2}-6}{-1}=6-3\sqrt{2}$

5. Algebraic surds

The same rules work when letters are involved. For example, $xx=x\sqrt{x}\sqrt{x}=x$ , as long as $\ge 0$ .

Common Mistake

Letters under roots

In IGCSE surd algebra, assume quantities under square roots are non-negative unless the question says otherwise.

Example

Algebraic conjugates

Simplify $(x+y)(x−y)(\sqrt{x}+\sqrt{y})(\sqrt{x}-\sqrt{y})$ .

Recognise the conjugate pair:
$(x+y)(x−y)=(x)2−(y)2(\sqrt{x}+\sqrt{y})(\sqrt{x}-\sqrt{y})=(\sqrt{x})^2-(\sqrt{y})^2$
Simplify each squared root:
$x - y$

Example

Squaring an algebraic surd bracket

Expand and simplify $(3p+q)2(3p+\sqrt{q})^2$ .

Write the bracket twice:
$(3p+q)2=(3p+q)(3p+q)(3p+\sqrt{q})^2=(3p+\sqrt{q})(3p+\sqrt{q})$
Expand all four terms:
$9p2+3pq+3pq+q9p^2+3p\sqrt{q}+3p\sqrt{q}+q$
Collect the middle terms:
$9p2+6pq+q9p^2+6p\sqrt{q}+q$

Exam technique

In the exam

Look for the largest square factor before simplifying a surd.
When expanding brackets, write all four products unless you are using a clear identity.
To rationalise, choose the multiplier carefully: same surd for one term, conjugate for two terms.
Finish by collecting like surds and checking the denominator has no surd left.

Self review

Check yourself

Can you simplify $98\sqrt{98}$ and $5205\sqrt{20}$ ?
Can you expand $(4−3)2(4-\sqrt{3})^2$ without losing the middle term?
Can you rationalise $82\frac{8}{\sqrt{2}}$ and $6+52+5\frac{6+\sqrt{5}}{2+\sqrt{5}}$ ?

Surds

What you'll learn

Recognise what a surd is and simplify square roots exactly.
Expand brackets involving surds and collect like terms.
Rationalise denominators, including when the denominator has two terms.
Use the same rules with algebraic surds such as $x\sqrt{x}$ .

1. What is a surd?

Before surds, you need to be confident with square numbers: 1, 4, 9, 16, 25, 36, 49, and so on.

A square root asks: “What number squares to give this?” For example, $36=6\sqrt{36}=6$ because $6^2=36$ .

Definition

Key vocabulary

A perfect square is a number made by squaring an integer, such as 16 or 81.
An irrational number cannot be written exactly as a fraction of integers.
A surd is an exact root that is irrational, such as $2\sqrt{2}$ , $7\sqrt{7}$ , or $45\sqrt{45}$ .

So $25\sqrt{25}$ is not a surd because it equals 5, but $5\sqrt{5}$ is a surd.

Example

Recognising surds

Decide which of $64\sqrt{64}$ , $18\sqrt{18}$ , and $10\sqrt{10}$ are surds.

First check for exact square roots:
$64=8\sqrt{64}=8$
Since $64\sqrt{64}$ is a whole number, it is not a surd.
The numbers 18 and 10 are not perfect squares, so $18\sqrt{18}$ and $10\sqrt{10}$ are surds.

2. Simplifying surds

To simplify a surd, look for the largest perfect-square factor inside the square root.

Key Idea

Pull out square factors

For non-negative values, you can split a product inside a square root: $ab=ab\sqrt{ab}=\sqrt{a}\sqrt{b}$ . This lets you take square factors out of the root.

For example, if a number contains a factor of 4, 9, 16, 25, 36, and so on, that factor can usually help you simplify.

Example

Writing a surd in simplest form

Write $72\sqrt{72}$ in the form $k2k\sqrt{2}$ , where $k$ is an integer.

Find a square factor of 72 that leaves 2:
$\times 2$
Split the square root and simplify the square part:
$72=36×2=362=62\sqrt{72}=\sqrt{36 \times 2}=\sqrt{36}\sqrt{2}=6\sqrt{2}$
Therefore the answer is $626\sqrt{2}$ .

Surds with a number in front

If there is already a number multiplying the surd, simplify the root first, then multiply the outside numbers.

Example

Simplifying with a coefficient

Write $4454\sqrt{45}$ in the form $k5k\sqrt{5}$ , where $k$ is an integer.

Simplify the surd part first:
$45=9×5=35\sqrt{45}=\sqrt{9 \times 5}=3\sqrt{5}$
Now multiply by the coefficient 4:
$445=4×35=1254\sqrt{45}=4 \times 3\sqrt{5}=12\sqrt{5}$
So $k = 12$ .

Common Mistake

Do not split sums inside roots

You may split multiplication, but not addition. For example, $9+16\sqrt{9+16}$ is not the same as $9+16\sqrt{9}+\sqrt{16}$ .

Like surds

Definition

Like surds

Like surds have the same root part, such as $373\sqrt{7}$ and $575\sqrt{7}$ . You can add or subtract like surds, just like collecting like terms in algebra.

For example, $23+73=932\sqrt{3}+7\sqrt{3}=9\sqrt{3}$ , but $23+752\sqrt{3}+7\sqrt{5}$ cannot be collected.

Example

Collecting like surds

Simplify $348+2753\sqrt{48}+2\sqrt{75}$ .

Simplify each surd separately:
$48=16×3=43\sqrt{48}=\sqrt{16 \times 3}=4\sqrt{3}$
Also simplify $75\sqrt{75}$ :
$75=25×3=53\sqrt{75}=\sqrt{25 \times 3}=5\sqrt{3}$
Substitute these back in:
$348+275=3(43)+2(53)3\sqrt{48}+2\sqrt{75}=3(4\sqrt{3})+2(5\sqrt{3})$
Collect the like surds:
$123+103=22312\sqrt{3}+10\sqrt{3}=22\sqrt{3}$

3. Expanding brackets with surds

Expand brackets with surds in the same way as algebra brackets: multiply every term in the first bracket by every term in the second bracket.

Remember that $aa=a\sqrt{a}\sqrt{a}=a$ . For example, $55=5\sqrt{5}\sqrt{5}=5$ .

Example

Expanding two brackets

Expand and simplify $(2+7)(3−7)(2+\sqrt{7})(3-\sqrt{7})$ .

Multiply out the four products:
$(2+7)(3−7)=6−27+37−(7)2(2+\sqrt{7})(3-\sqrt{7})=6-2\sqrt{7}+3\sqrt{7}-(\sqrt{7})^2$
Simplify the square root squared:
$6−27+37−76-2\sqrt{7}+3\sqrt{7}-7$
Collect the ordinary numbers and the like surds:
$−1+7-1+\sqrt{7}$

Conjugates

Definition

Conjugates

Two expressions such as $a+ba+\sqrt{b}$ and $a−ba-\sqrt{b}$ are called conjugates. Their product has no surd part because the middle terms cancel.

This is just the difference of two squares:

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

Example

Using conjugates

Expand and simplify $(5+6)(5−6)(5+\sqrt{6})(5-\sqrt{6})$ .

Recognise the brackets as conjugates.
Use the difference of two squares:
$(5+6)(5−6)=52−(6)2(5+\sqrt{6})(5-\sqrt{6})=5^2-(\sqrt{6})^2$
Simplify:
$25 - 6 = 19$

Squaring a bracket

When you square a bracket, write it twice. This helps you avoid missing the middle term.

Example

Squaring a surd bracket

Write $(3−5)2(3-\sqrt{5})^2$ in the form $a+b5a+b\sqrt{5}$ .

Write the squared bracket as two identical brackets:
$(3−5)2=(3−5)(3−5)(3-\sqrt{5})^2=(3-\sqrt{5})(3-\sqrt{5})$
Expand:
$9−35−35+59-3\sqrt{5}-3\sqrt{5}+5$
Collect terms:
$14−6514-6\sqrt{5}$

Common Mistake

Missing the middle term

$(3−5)2(3-\sqrt{5})^2$ is not just $32+(5)23^2+(\sqrt{5})^2$ . The two middle terms are essential.

4. Rationalising the denominator

Definition

Rationalising the denominator

To rationalise the denominator means rewriting a fraction so there is no surd in the denominator.

A fraction is not considered fully simplified if the denominator still contains a surd.

Single surd denominator

If the denominator is just one square root, multiply the numerator and denominator by that square root.

Example

Rationalising a single surd denominator

Rationalise $123\frac{12}{\sqrt{3}}$ .

Multiply top and bottom by $3\sqrt{3}$ :
$123×33=1233\frac{12}{\sqrt{3}}\times\frac{\sqrt{3}}{\sqrt{3}}=\frac{12\sqrt{3}}{3}$
Simplify the fraction:
$1233=43\frac{12\sqrt{3}}{3}=4\sqrt{3}$

If the numerator has more than one term, multiply the whole numerator by the surd.

Example

Rationalising with a bracket on top

Simplify fully $4+62\frac{4+\sqrt{6}}{\sqrt{2}}$ .

Multiply the numerator and denominator by $2\sqrt{2}$ :
$4+62×22=42+122\frac{4+\sqrt{6}}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{4\sqrt{2}+\sqrt{12}}{2}$
Simplify $12\sqrt{12}$ :
$12=23\sqrt{12}=2\sqrt{3}$
Divide both terms in the numerator by 2:
$42+232=22+3\frac{4\sqrt{2}+2\sqrt{3}}{2}=2\sqrt{2}+\sqrt{3}$

Two-term denominator

If the denominator is something like $3+23+\sqrt{2}$ , multiply by its conjugate $3−23-\sqrt{2}$ .

Tip

Choosing the multiplier

For a one-term denominator, multiply by the same surd. For a two-term denominator, multiply by the conjugate.

Example

Rationalising a two-term denominator

Show that $10+23+2\frac{10+\sqrt{2}}{3+\sqrt{2}}$ simplifies to $4−24-\sqrt{2}$ .

Multiply top and bottom by the conjugate of the denominator:
$10+23+2×3−23−2\frac{10+\sqrt{2}}{3+\sqrt{2}}\times\frac{3-\sqrt{2}}{3-\sqrt{2}}$
The denominator becomes a difference of two squares:
$(3+2)(3−2)=9−2=7(3+\sqrt{2})(3-\sqrt{2})=9-2=7$
Expand the numerator:
$(10+2)(3−2)=30−102+32−2=28−72(10+\sqrt{2})(3-\sqrt{2})=30-10\sqrt{2}+3\sqrt{2}-2=28-7\sqrt{2}$
Divide by 7:
$28−727=4−2\frac{28-7\sqrt{2}}{7}=4-\sqrt{2}$

Common Mistake

Only changing the denominator

You must multiply the numerator and denominator by the same expression. Otherwise you change the value of the fraction.

Fractions inside fractions

Sometimes the denominator contains a fraction, such as $12+1\frac{1}{\sqrt{2}}+1$ . First combine the denominator, then rationalise if needed.

Example

Simplifying a fraction inside a fraction

Simplify $312+1\frac{3}{\frac{1}{\sqrt{2}}+1}$ .

Write the denominator as one fraction:
$12+1=1+22\frac{1}{\sqrt{2}}+1=\frac{1+\sqrt{2}}{\sqrt{2}}$
Divide by this fraction by multiplying by its reciprocal:
$31+22=321+2\frac{3}{\frac{1+\sqrt{2}}{\sqrt{2}}}=\frac{3\sqrt{2}}{1+\sqrt{2}}$
Rationalise using the conjugate $1−21-\sqrt{2}$ :
$321+2×1−21−2=32−6−1=6−32\frac{3\sqrt{2}}{1+\sqrt{2}}\times\frac{1-\sqrt{2}}{1-\sqrt{2}}=\frac{3\sqrt{2}-6}{-1}=6-3\sqrt{2}$

5. Algebraic surds

The same rules work when letters are involved. For example, $xx=x\sqrt{x}\sqrt{x}=x$ , as long as $\ge 0$ .

Common Mistake

Letters under roots

In IGCSE surd algebra, assume quantities under square roots are non-negative unless the question says otherwise.

Example

Algebraic conjugates

Simplify $(x+y)(x−y)(\sqrt{x}+\sqrt{y})(\sqrt{x}-\sqrt{y})$ .

Recognise the conjugate pair:
$(x+y)(x−y)=(x)2−(y)2(\sqrt{x}+\sqrt{y})(\sqrt{x}-\sqrt{y})=(\sqrt{x})^2-(\sqrt{y})^2$
Simplify each squared root:
$x - y$

Example

Squaring an algebraic surd bracket

Expand and simplify $(3p+q)2(3p+\sqrt{q})^2$ .

Write the bracket twice:
$(3p+q)2=(3p+q)(3p+q)(3p+\sqrt{q})^2=(3p+\sqrt{q})(3p+\sqrt{q})$
Expand all four terms:
$9p2+3pq+3pq+q9p^2+3p\sqrt{q}+3p\sqrt{q}+q$
Collect the middle terms:
$9p2+6pq+q9p^2+6p\sqrt{q}+q$

Exam technique

In the exam

Look for the largest square factor before simplifying a surd.
When expanding brackets, write all four products unless you are using a clear identity.
To rationalise, choose the multiplier carefully: same surd for one term, conjugate for two terms.
Finish by collecting like surds and checking the denominator has no surd left.

Self review

Check yourself

Can you simplify $98\sqrt{98}$ and $5205\sqrt{20}$ ?
Can you expand $(4−3)2(4-\sqrt{3})^2$ without losing the middle term?
Can you rationalise $82\frac{8}{\sqrt{2}}$ and $6+52+5\frac{6+\sqrt{5}}{2+\sqrt{5}}$ ?

Surds

What you'll learn

1. What is a surd?

2. Simplifying surds

Surds with a number in front

Like surds

3. Expanding brackets with surds

Conjugates

Squaring a bracket

4. Rationalising the denominator

Single surd denominator

Two-term denominator

Fractions inside fractions

5. Algebraic surds

Recap questions

You've reached the end

Surds

What you'll learn

1. What is a surd?

2. Simplifying surds

Surds with a number in front

Like surds

3. Expanding brackets with surds

Conjugates

Squaring a bracket

4. Rationalising the denominator

Single surd denominator

Two-term denominator

Fractions inside fractions

5. Algebraic surds

Recap questions

You've reached the end

Surds

What you'll learn

1. What is a surd?

2. Simplifying surds

Surds with a number in front

Like surds

3. Expanding brackets with surds

Conjugates

Squaring a bracket

4. Rationalising the denominator

Single surd denominator

Two-term denominator

Fractions inside fractions

5. Algebraic surds

Recap questions

You've reached the end

Surds Revision Guide

Surds

What you'll learn

1. What is a surd?

2. Simplifying surds

Surds with a number in front

Like surds

3. Expanding brackets with surds

Conjugates

Squaring a bracket

4. Rationalising the denominator

Single surd denominator

Two-term denominator

Fractions inside fractions

5. Algebraic surds

Recap questions

You've reached the end

Surds Revision Guide