Adding and Subtracting Surds
Only like surds can be added or subtracted, in the same way as like terms: \(3\sqrt{5} + \sqrt{5} = 4\sqrt{5}\). In NSW Year 9 Advanced maths, each surd is simplified first so that hidden like surds appear (NESA outcome MA5‑IND‑P‑02).
You will simplify sums such as \(\sqrt{45} + \sqrt{80}\) and \(2\sqrt{50} - 3\sqrt{18} + \sqrt{3}\), leave two different surds in one answer, and work with surds that contain a pronumeral, like \(\sqrt{9x}\). You will explain why \(\sqrt{2} + \sqrt{8}\) is not \(\sqrt{10}\), find a missing number in a surd equation, and give exact perimeters of a rectangle and a triangle with surd sides. It is part of Year 9 Advanced Pythagoras Theorem.
Every question with a fully worked solution.
- Adding and Subtracting Surds - Video - Adding and subtracting surds Watch
Theory
Only like surds can be added or subtracted: \(4\sqrt{3} + 2\sqrt{3} = 6\sqrt{3}\). This Year 9 Advanced maths page shows how to simplify each surd first so that like surds appear, keep unlike surds apart, and use the skill for perimeters and equations with a missing number.
Like surds have the same number under the root, such as \(4\sqrt{3}\) and \(-\sqrt{3}\). Unlike surds have different numbers under the root, such as \(\sqrt{3}\) and \(\sqrt{5}\).
Like surds are added and subtracted the way like terms are in algebra. \(4\sqrt{3} + 2\sqrt{3}\) works like \(4x + 2x\): the answer is \(6\sqrt{3}\). Unlike surds cannot be combined, just as \(4x + 2y\) cannot.
Surds that look unlike can become like once they are simplified. \(\sqrt{8}\) and \(\sqrt{18}\) look different, but \(\sqrt{8} = 2\sqrt{2}\) and \(\sqrt{18} = 3\sqrt{2}\). So always simplify every surd before collecting.
The answer is in simplest form when each surd is simplified and like surds are collected. An answer can have more than one term, such as \(\sqrt{7} + 5\sqrt{5}\).
Like surds have the same number under the root. Add or subtract the numbers in front and keep the surd:
There is no rule for adding roots of different numbers:
How to add and subtract surds
- Simplify every surd first, multiplying by any number already in front.
- Group like surds: the terms with the same number under the root.
- Add or subtract the numbers in front of each group. Keep the surd unchanged.
- Leave unlike surds as separate terms. A whole number and a surd also stay separate.
- For an unknown, collect the known surds on one side, then write the unknown surd as an entire surd.
None of these are like surds yet. Simplify each one, keeping the number in front.
| \(3\sqrt{12}\) | \(=\) | \(3 \times 2\sqrt{3} = 6\sqrt{3}\) |
| \(\sqrt{75}\) | \(=\) | \(\sqrt{25} \times \sqrt{3} = 5\sqrt{3}\) |
| \(\sqrt{27}\) | \(=\) | \(\sqrt{9} \times \sqrt{3} = 3\sqrt{3}\) |
Now all three are multiples of \(\sqrt{3}\). Collect the numbers in front.
| \(3\sqrt{12} - \sqrt{75} + \sqrt{27}\) | \(=\) | \(6\sqrt{3} - 5\sqrt{3} + 3\sqrt{3}\) |
| \(\) | \(=\) | \((6 - 5 + 3)\sqrt{3}\) |
| \(\) | \(=\) | \(\mathbf{4\sqrt{3}}\) |
| \(\sqrt{63}\) | \(=\) | \(\sqrt{9} \times \sqrt{7} = 3\sqrt{7}\) |
| \(\sqrt{20}\) | \(=\) | \(\sqrt{4} \times \sqrt{5} = 2\sqrt{5}\) |
| \(\sqrt{45}\) | \(=\) | \(\sqrt{9} \times \sqrt{5} = 3\sqrt{5}\) |
Group the \(\sqrt{7}\) terms and the \(\sqrt{5}\) terms separately, like \(x\) and \(y\) terms in algebra.
| \(\) | \(\) | \(3\sqrt{7} + 2\sqrt{5} - 2\sqrt{7} + 3\sqrt{5}\) |
| \(\) | \(=\) | \((3 - 2)\sqrt{7} + (2 + 3)\sqrt{5}\) |
| \(\) | \(=\) | \(\mathbf{\sqrt{7} + 5\sqrt{5}}\) |
\(\sqrt{7}\) and \(5\sqrt{5}\) are unlike surds, so the answer has two terms. Write \(\sqrt{7}\), not \(1\sqrt{7}\).
(i) Simplify \(\dfrac{\sqrt{48}}{2} - \dfrac{\sqrt{27}}{3}\).
(ii) Find \(m\) if \(\sqrt{m} - \sqrt{28} = \sqrt{7}\).
(i) Simplify each surd, then cancel inside each fraction.
| \(\dfrac{\sqrt{48}}{2}\) | \(=\) | \(\dfrac{4\sqrt{3}}{2} = 2\sqrt{3}\) |
| \(\dfrac{\sqrt{27}}{3}\) | \(=\) | \(\dfrac{3\sqrt{3}}{3} = \sqrt{3}\) |
| \(2\sqrt{3} - \sqrt{3}\) | \(=\) | \(\mathbf{\sqrt{3}}\) |
(ii) Simplify \(\sqrt{28} = 2\sqrt{7}\), get \(\sqrt{m}\) on its own, then write it as an entire surd.
| \(\sqrt{m} - 2\sqrt{7}\) | \(=\) | \(\sqrt{7}\) |
| \(\sqrt{m}\) | \(=\) | \(3\sqrt{7}\) |
| \(\) | \(=\) | \(\sqrt{9 \times 7} = \sqrt{63}\) |
| \(m\) | \(=\) | \(\mathbf{63}\) |
Check: \(\sqrt{63} - \sqrt{28} = 3\sqrt{7} - 2\sqrt{7} = \sqrt{7}\).
(i) Find its exact perimeter in simplest surd form.
(ii) Is it right-angled?
(i) Simplify each side. All three become multiples of \(\sqrt{3}\).
| \(P\) | \(=\) | \(\sqrt{12} + \sqrt{27} + \sqrt{48}\) |
| \(\) | \(=\) | \(2\sqrt{3} + 3\sqrt{3} + 4\sqrt{3}\) |
| \(\) | \(=\) | \(\mathbf{9\sqrt{3}}\ \text{cm}\) |
(ii) Square each side: \((\sqrt{12})^2 = 12\), and so on. Compare the two smaller squares with the largest.
| \(12 + 27\) | \(=\) | \(39\) |
| \((\sqrt{48})^2\) | \(=\) | \(48\) |
\(39 \ne 48\), so the triangle is not right-angled.
Common pitfalls
Frequently asked questions
How do you add surds?
Simplify each surd first. Then add the numbers in front of like surds, the ones with the same number under the root, and keep the surd. For example, \(\sqrt{12} + \sqrt{27} = 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}\).
What are like surds?
Like surds have the same number under the root once they are simplified, such as \(2\sqrt{3}\) and \(7\sqrt{3}\). They can be added and subtracted.
Can you add root 2 and root 3?
No. They are unlike surds, so \(\sqrt{2} + \sqrt{3}\) stays as it is. It is about \(3.15\), while \(\sqrt{5}\) is only about \(2.24\).
Why do you simplify surds before adding them?
Simplifying can turn unlike-looking surds into like surds. \(\sqrt{32}\) and \(\sqrt{50}\) look different, but they are \(4\sqrt{2}\) and \(5\sqrt{2}\), so together they make \(9\sqrt{2}\).
Is the square root of a plus b the same as root a plus root b?
No. For example, \(\sqrt{9} + \sqrt{16} = 3 + 4 = 7\), but \(\sqrt{25} = 5\).