MISSION · Number sense
Simplifying Radicals
Getting the square numbers out from under the root
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
A radical is in simplest form when nothing square is left under the root sign — and getting there means splitting the number into a square factor and whatever’s left.
The splitting rule
\[\sqrt{ab} = \sqrt{a} \times \sqrt{b}\]Roots split across multiplication. So look for the largest square factor and pull it out:
\[\sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}\]Choosing the largest square matters for efficiency, not correctness. Using \(\sqrt{4 \times 18}\) gives \(2\sqrt{18}\), which isn’t finished — 18 still contains a 9 — so you simplify twice instead of once.
The rule does not work across addition. \(\sqrt{9 + 16} = \sqrt{25} = 5\), not \(3 + 4\). This is the same shape of error as expanding \((a+b)^2\) wrongly, and it comes from the same instinct.
Why bother
Because \(6\sqrt{2}\) is exact and 8.485 is not. Keeping surds carries the precise value through a calculation, so rounding happens once at the end rather than accumulating at every step.
It also reveals structure that decimals hide. \(\sqrt{8} + \sqrt{18}\) looks like two unrelated numbers until you simplify: \(2\sqrt{2} + 3\sqrt{2} = 5\sqrt{2}\). In decimals that’s 2.828 + 4.243 = 7.071, which tells you nothing about where it came from.
Adding needs like radicals
Surds add the way like terms do in algebra — only when the part under the root matches.
\[3\sqrt{5} + 4\sqrt{5} = 7\sqrt{5}\]but \(3\sqrt{5} + 4\sqrt{2}\) cannot be combined at all. Treat \(\sqrt{5}\) as though it were \(x\): you can collect \(3x + 4x\), and you can’t collect \(3x + 4y\).
Always simplify first, because radicals that look unlike often aren’t. \(\sqrt{12}\) and \(\sqrt{27}\) become \(2\sqrt{3}\) and \(3\sqrt{3}\), which add to \(5\sqrt{3}\).
Rationalising the denominator
Convention says no radical is left on the bottom of a fraction. To clear one, multiply top and bottom by it:
\[\frac{3}{\sqrt{2}} = \frac{3}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{3\sqrt{2}}{2}\]Multiplying by \(\frac{\sqrt{2}}{\sqrt{2}}\) is multiplying by 1, so the value is untouched — only its appearance changes.
For a two-term denominator like \(1 + \sqrt{3}\), multiply by its conjugate, \(1 - \sqrt{3}\). The difference-of-squares pattern kills the middle terms and leaves \(1 - 3 = -2\), with no radical in sight.
The convention dates from the days of hand calculation, when dividing by 1.414 was much worse than dividing by 2. It survives because it gives everyone one agreed form to write an answer in.
Common mix-ups
- Splitting a root across addition. Multiplication only.
- Adding unlike radicals. \(\sqrt{2} + \sqrt{3}\) stays as it is.
- Stopping before simplest form, leaving a square factor underneath.
- Multiplying only the denominator when rationalising. Both parts, or the value changes.
Check yourself
Simplify \(\sqrt{50} + \sqrt{32}\), and rationalise \(\frac{4}{\sqrt{3}}\).
\(5\sqrt{2} + 4\sqrt{2} = 9\sqrt{2}\). And \(\frac{4}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{4\sqrt{3}}{3}\).
LEVEL 3 · REFLECT
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