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Number · Surds

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Simplifying surds

Why surds are kept as exact values, why √(ab) = √a × √b lets you pull square factors out, and how to collect like surds — and why √a + √b is not √(a+b).

Number · Surds

Simplifying surds

Why surds are kept as exact values, why √(ab) = √a × √b lets you pull square factors out, and how to collect like surds — and why √a + √b is not √(a+b).

Why it works

A surd is a root that isn't a whole number — 2\sqrt{2}, 48\sqrt{48}, 535\sqrt{3}. We keep them as symbols instead of decimals because they are exact: 2\sqrt{2} squared is exactly 22, while 1.4142=1.9993961.414^2 = 1.999396 — close, but wrong. When a question says "give an exact answer" or "in the form aba\sqrt{b}", a rounded decimal scores nothing.

The one rule everything comes from:

ab=a×b.\sqrt{ab} = \sqrt{a} \times \sqrt{b}.

Why? Because (ab)2=a a b b=ab(\sqrt{a}\sqrt{b})^2 = \sqrt{a}\,\sqrt{a}\,\sqrt{b}\,\sqrt{b} = ab — so ab\sqrt{a}\sqrt{b} is the positive number that squares to abab, which is exactly what ab\sqrt{ab} means. In particular a×a=a\sqrt{a} \times \sqrt{a} = a: the root and the square cancel exactly.

Simplifying means pulling out square factors. To simplify 48\sqrt{48}, find the largest square factor of 48: 48=16×348 = 16 \times 3, so

48=16×3=43.\sqrt{48} = \sqrt{16}\times\sqrt{3} = 4\sqrt{3}.

If you spot only a smaller square factor — 48=412=212\sqrt{48} = \sqrt{4}\sqrt{12} = 2\sqrt{12} — you're not wrong, just not finished: 12\sqrt{12} still hides a square factor. Keep going until the number under the root has no square factor left.

Like surds collect; unlike surds don't. 53+23=735\sqrt{3} + 2\sqrt{3} = 7\sqrt{3} for the same reason 5x+2x=7x5x + 2x = 7x — the 3\sqrt{3} is a common factor. The root part stays put: 23+332\sqrt{3} + 3\sqrt{3} is 535\sqrt{3}, not 565\sqrt{6}. And 3+5\sqrt{3} + \sqrt{5} cannot be combined at all — often the whole point of a question is to make surds alike by simplifying first: 12+27=23+33=53\sqrt{12} + \sqrt{27} = 2\sqrt{3} + 3\sqrt{3} = 5\sqrt{3}.

The rule that does NOT exist: a+b≠a+b\sqrt{a} + \sqrt{b} \ne \sqrt{a + b}. Test it: 9+16=3+4=7\sqrt{9} + \sqrt{16} = 3 + 4 = 7, but 9+16=25=5\sqrt{9 + 16} = \sqrt{25} = 5. Roots split over multiplication, never over addition.