A surd is the square root of a natural number that is irrational, such as square root of two or square root of seven. Roots of perfect squares like square root of nine equals three are not surds.

How do you simplify a surd like the square root of 72?

Find the largest perfect square factor. Seventy two is thirty six times two, so the square root of seventy two is the square root of thirty six times the square root of two, which is six root two.

Is the square root of a plus b the same as root a plus root b?

No. The root does not split over a sum. The square root of nine plus sixteen is the square root of twenty five which is five, but root nine plus root sixteen is three plus four which is seven.

A surd is the square root of a natural number that is irrational, such as square root of two or square root of seven. Roots of perfect squares like square root of nine equals three are not surds.

How do you simplify a surd like the square root of 72?

Find the largest perfect square factor. Seventy two is thirty six times two, so the square root of seventy two is the square root of thirty six times the square root of two, which is six root two.

Is the square root of a plus b the same as root a plus root b?

No. The root does not split over a sum. The square root of nine plus sixteen is the square root of twenty five which is five, but root nine plus root sixteen is three plus four which is seven.

Year 11 Methods (Unit 1 & 2) Quadratics

Simplifying Surds

Q: When can you add or subtract surds?

Only when they are like surds, meaning the same number under the root. Simplify each surd first, then collect the coefficients. For example five root two plus two root two is seven root two.

Q: How do you multiply two surds?

Multiply the coefficients together and multiply the numbers under the roots together, then simplify. So p root a times q root b equals p q root a b.

20 practice questions 0 video lessons Theory + worked examples

Master simplifying surds for Queensland Year 11 Mathematical Methods (QCAA). A surd is an irrational root, and simplifying it means writing that root in its simplest exact form by pulling out perfect-square factors.

You will learn to break a root into factors, extract perfect squares to reach simplest form, recognise like surds, and add, subtract, multiply and divide surds confidently — building the exact-value skills used throughout algebra.

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Theory

In Year 11 Mathematical Methods (QCAA, Unit 1), a surd is an irrational number written with a root sign, such as \(\sqrt{2}\) or \(\sqrt{5}\). Simplifying surds means writing a square root in its simplest exact form by extracting perfect-square factors, then adding, subtracting, multiplying and dividing surds. This page covers \(\sqrt{ab}=\sqrt{a}\,\sqrt{b}\), like surds, and surd arithmetic.

A surd is a square root of a natural number that is irrational — it cannot be written as an exact fraction or decimal. \(\sqrt{2},\ \sqrt{3}\) and \(\sqrt{5}\) are surds, but \(\sqrt{9}=3\) and \(\sqrt{16}=4\) are not, because \(9\) and \(16\) are perfect squares.

A surd is in simplest form when the number under the root has no perfect-square factor larger than \(1\). We simplify by finding the largest perfect-square factor and splitting the root with \(\sqrt{ab}=\sqrt{a}\,\sqrt{b}\).

Like surds have the same number under the root (for example \(3\sqrt{2}\) and \(5\sqrt{2}\)). Only like surds can be added or subtracted, by collecting their coefficients.

Simplify first, then combine. Always reduce each surd to simplest form before deciding which terms are like surds — \(\sqrt{8}\) and \(\sqrt{2}\) look different but \(\sqrt{8}=2\sqrt{2}\).

A square of area \(8\) has side \(\sqrt{8}=2\sqrt{2}\) — the perfect-square factor \(4\) comes out as \(2\).

\(\sqrt{8}=2\sqrt{2}\approx 2.83\) lies between \(2\) and \(3\), since \(4<8<9\).

The product and quotient rules for surds (for \(a,b\ge 0\)):

\[\sqrt{ab}=\sqrt{a}\,\sqrt{b}\]

\sqrt{a b} = \sqrt{a} \sqrt{b}

\[\dfrac{\sqrt{a}}{\sqrt{b}}=\sqrt{\dfrac{a}{b}}\]

\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}

Collecting like surds (same number under the root):

\[p\sqrt{n}\pm q\sqrt{n}=(p\pm q)\sqrt{n}\]

p \sqrt{n} \pm q \sqrt{n} = (p \pm q) \sqrt{n}

Multiplying: \(p\sqrt{a}\times q\sqrt{b}=pq\sqrt{ab}\). Multiply the coefficients, multiply the numbers under the roots, then simplify the surd.

How to simplify a surd

Factor: find the largest perfect-square factor of the number under the root (\(4,9,16,25,\dots\)).
Split: use \(\sqrt{ab}=\sqrt{a}\,\sqrt{b}\) and take the exact root of the perfect square.
Combine: for sums, collect like surds; for products, multiply coefficients and roots (\(p\sqrt{a}\times q\sqrt{b}=pq\sqrt{ab}\)), then re-simplify.

Example 1 — Simplify a single surd

Write \(\sqrt{72}\) in simplest form.

Solution

Largest perfect-square factor — \(72=36\times 2\):

\(\sqrt{72}\)

\(=\)

\(\sqrt{36\times 2}\)

Split the root and take \(\sqrt{36}\):

	\(=\)	\(\sqrt{36}\,\sqrt{2}\)
	\(=\)	\(6\sqrt{2}\)

\(\sqrt{72}=6\sqrt{2}\).

\sqrt{72} = 6 \sqrt{2}

Example 2 — Add and subtract like surds

Simplify \(\sqrt{50}-\sqrt{18}+\sqrt{8}\).

Solution

Simplify each surd first:

\(\sqrt{50}\)	\(=\)	\(\sqrt{25\times 2}=5\sqrt{2}\)
\(\sqrt{18}\)	\(=\)	\(\sqrt{9\times 2}=3\sqrt{2}\)
\(\sqrt{8}\)	\(=\)	\(\sqrt{4\times 2}=2\sqrt{2}\)

They are all like surds — collect the coefficients:

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

\(\sqrt{50}-\sqrt{18}+\sqrt{8}=4\sqrt{2}\).

4 \sqrt{2}

Example 3 — Multiply surds

Simplify \(2\sqrt{6}\times 5\sqrt{3}\).

Solution

Multiply coefficients, and the numbers under the roots:

\(2\sqrt{6}\times 5\sqrt{3}\)	\(=\)	\((2\times 5)\sqrt{6\times 3}\)
	\(=\)	\(10\sqrt{18}\)

Simplify the surd — \(18=9\times 2\):

\(10\sqrt{18}\)	\(=\)	\(10\sqrt{9}\,\sqrt{2}\)
	\(=\)	\(10\times 3\sqrt{2}\)
	\(=\)	\(30\sqrt{2}\)

\(2\sqrt{6}\times 5\sqrt{3}=30\sqrt{2}\).

30 \sqrt{2}

Example 4 — Expand and simplify

Expand and simplify \(\sqrt{5}\,\bigl(2\sqrt{5}-\sqrt{15}\bigr)\).

Solution

Distribute \(\sqrt{5}\) over the bracket:

\(\sqrt{5}\cdot 2\sqrt{5}\)	\(=\)	\(2\sqrt{25}=2\times 5=10\)
\(\sqrt{5}\cdot \sqrt{15}\)	\(=\)	\(\sqrt{75}=\sqrt{25\times 3}=5\sqrt{3}\)

Combine the two terms:

\(\sqrt{5}\,(2\sqrt{5}-\sqrt{15})\)

\(=\)

\(10-5\sqrt{3}\)

\(\sqrt{5}\,\bigl(2\sqrt{5}-\sqrt{15}\bigr)=10-5\sqrt{3}\).

10 - 5 \sqrt{3}

Common pitfalls

Adding the numbers under the roots. \(\sqrt{a}+\sqrt{b}\ne \sqrt{a+b}\). For example \(\sqrt{9}+\sqrt{16}=3+4=7\), not \(\sqrt{25}=5\).

Not extracting the largest square. Writing \(\sqrt{72}=2\sqrt{18}\) is not finished — \(18\) still has the factor \(9\). Use the largest perfect square, \(36\).

Combining unlike surds. \(3\sqrt{2}+4\sqrt{3}\) cannot be simplified; only surds with the same number under the root are like surds.

Frequently asked questions

What is a surd?

A surd is the square root of a natural number that is irrational, such as \(\sqrt{2}\) or \(\sqrt{7}\). Roots of perfect squares like \(\sqrt{9}=3\) are not surds.

How do you simplify a surd like \(\sqrt{72}\)?

Find the largest perfect-square factor: \(72=36\times 2\). Then \(\sqrt{72}=\sqrt{36}\,\sqrt{2}=6\sqrt{2}\).

When can you add or subtract surds?

Only when they are like surds — the same number under the root. Simplify each surd first, then collect coefficients, e.g. \(5\sqrt{2}+2\sqrt{2}=7\sqrt{2}\).

How do you multiply two surds?

Multiply the coefficients and multiply the numbers under the roots: \(p\sqrt{a}\times q\sqrt{b}=pq\sqrt{ab}\), then simplify the result.

Is \(\sqrt{a+b}\) the same as \(\sqrt{a}+\sqrt{b}\)?

No. The root does not split over a sum. For instance \(\sqrt{9+16}=\sqrt{25}=5\), but \(\sqrt{9}+\sqrt{16}=3+4=7\).

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Rationalising Denominators

Simplifying Surds

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How to simplify a surd

Common pitfalls

Frequently asked questions

Theory