How to Add, Subtract, Multiply and Divide Surds

Master the four operations with surds for Edexcel IGCSE Maths: simplify a surd, add and subtract like surds, and multiply and divide surds, always leaving the answer in simplest surd form. Every worked example shows the aligned steps and the slips to avoid.

When you are ready, try the auto-marked, randomly generated practice questions below for instant feedback and exam-style revision.

Prior Knowledge You should be comfortable finding square factors from prime factorisation, and collecting like terms from simplifying algebraic expressions.
⇩ Jump to Practice Questions ⇩

Working with Surds: the Four Operations

A surd is a root that stays irrational, such as \(\sqrt{2}\) or \(\sqrt{5}\); it is an exact value. To add, subtract, multiply or divide surds you use just three ideas, then always leave the answer in simplest surd form (no square factor left under the root).

Like surds only
\(a\sqrt{k} \pm b\sqrt{k}=(a\pm b)\sqrt{k}\)
Add or subtract only when the number under the root matches, exactly like collecting like terms.
Product rule
\(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\)
Multiply the numbers in front, multiply the radicands, then take out any square factor.
Quotient rule
\(\dfrac{\sqrt{a}}{\sqrt{b}}=\sqrt{\dfrac{a}{b}}\)
Divide the numbers in front, divide the radicands, then simplify.
Simplest form
\(\sqrt{ab^2}=b\sqrt{a}\)
Take out the largest square factor so nothing left under the root is a perfect square.

Worked Examples

💡 Simplify a surd

Simplify \(\sqrt{98}\).

\[ \begin{array}{rcl} \sqrt{98} &=& \sqrt{49\times 2} \\ &=& \sqrt{49}\,\sqrt{2} \\ &=& 7\sqrt{2} \end{array} \]
What is happening?
  1. Find the largest square factor: \(98=49\times 2\).
  2. Split the root using the product rule.
  3. \(\sqrt{49}=7\), so the surd becomes \(7\sqrt{2}\).

💡 Add like surds

Simplify \(2\sqrt{20}+\sqrt{45}\).

\[ \begin{array}{rcl} 2\sqrt{20}+\sqrt{45} &=& 4\sqrt{5}+3\sqrt{5} \\ &=& (4+3)\sqrt{5} \\ &=& 7\sqrt{5} \end{array} \]
What is happening?
  1. Simplify each surd first: \(2\sqrt{20}=4\sqrt{5}\) and \(\sqrt{45}=3\sqrt{5}\).
  2. Now they are like surds (both \(\sqrt{5}\)).
  3. Add the numbers in front to get \(7\sqrt{5}\).

💡 Subtract like surds

Simplify \(3\sqrt{8}-\sqrt{50}\).

\[ \begin{array}{rcl} 3\sqrt{8}-\sqrt{50} &=& 6\sqrt{2}-5\sqrt{2} \\ &=& (6-5)\sqrt{2} \\ &=& \sqrt{2} \end{array} \]
What is happening?
  1. Simplify: \(3\sqrt{8}=6\sqrt{2}\) and \(\sqrt{50}=5\sqrt{2}\).
  2. Both are \(\sqrt{2}\), so subtract the coefficients.
  3. \(6-5=1\), so the answer is just \(\sqrt{2}\).

💡 Multiply surds

Simplify \(2\sqrt{6}\times\sqrt{3}\).

\[ \begin{array}{rcl} 2\sqrt{6}\times\sqrt{3} &=& 2\sqrt{18} \\ &=& 2\times 3\sqrt{2} \\ &=& 6\sqrt{2} \end{array} \]
What is happening?
  1. Multiply radicands: \(\sqrt{6}\times\sqrt{3}=\sqrt{18}\).
  2. Simplify \(\sqrt{18}=3\sqrt{2}\).
  3. Multiply the numbers in front: \(2\times 3=6\).

💡 When the surd cancels

Simplify \(4\sqrt{3}\times 2\sqrt{3}\).

\[ \begin{array}{rcl} 4\sqrt{3}\times 2\sqrt{3} &=& 8\sqrt{9} \\ &=& 8\times 3 \\ &=& 24 \end{array} \]
What is happening?
  1. \(\sqrt{3}\times\sqrt{3}=3\), so the root disappears.
  2. Multiply the coefficients: \(4\times 2=8\).
  3. The exact answer is the whole number \(24\).

💡 Divide surds

Simplify \(\dfrac{6\sqrt{40}}{2\sqrt{5}}\).

\[ \begin{array}{rcl} \dfrac{6\sqrt{40}}{2\sqrt{5}} &=& \dfrac{6}{2}\,\sqrt{\dfrac{40}{5}} \\ &=& 3\sqrt{8} \\ &=& 3\times 2\sqrt{2} \\ &=& 6\sqrt{2} \end{array} \]
What is happening?
  1. Divide the numbers in front and divide the radicands.
  2. \(\dfrac{40}{5}=8\), giving \(3\sqrt{8}\).
  3. Simplify \(\sqrt{8}=2\sqrt{2}\) to reach \(6\sqrt{2}\).

🔑Key Points

  • Simplify every surd first, then combine only like surds.
  • \(\sqrt{a}\times\sqrt{b}=\sqrt{ab}\); take out square factors at the end.
  • \(\sqrt{a}\times\sqrt{a}=a\), so a product or quotient can become a whole number.
  • Keep the exact surd form unless the question asks for a decimal.

⚠️Common Pitfalls

  • Leaving a square factor under the root: write \(2\sqrt{2}\), not \(\sqrt{8}\).
  • Trying to add unlike surds: \(\sqrt{2}+\sqrt{3}\) does not combine.
  • Writing \(\sqrt{a}+\sqrt{b}=\sqrt{a+b}\); that rule is false.
  • Stopping too early on a division that actually cancels to a whole number.
⇩ Jump to Practice Questions ⇩

Once you can add, subtract, multiply and divide surds, the next step is clearing surds from the bottom of a fraction.

Next: Simplifying & Rationalising Surds

Operations with Surds: Practice Rooms

These operations with surds practice rooms give you unlimited, auto-marked questions for Edexcel IGCSE Maths: simplify a surd, add and subtract like surds, and multiply and divide surds, with every answer checked in simplest surd form. Room 1 takes out square factors; Room 2 combines like surds (simplify first where needed); Room 3 multiplies and divides; Room 4 mixes every type. Difficulty rises from Starter to Master left to right across each grid. Type a surd with the √ button or as sqrt(2); a coefficient goes in front (3√2); use / for a fraction (√2/3) and a leading minus for a negative (-2√5). If the surds cancel, type the whole number or fraction. Give every answer fully simplified: an un-simplified but equal form such as √8 for 2√2 is marked wrong.

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Input tips: 3√2, 3\sqrt{2} and 3sqrt(2) are all accepted; a fraction coefficient may be written √2/3 or (1/3)√2. A radicand with a square factor left in (for example √12 instead of 2√3) is marked wrong on purpose: simplest surd form is the skill.