Not every expression written with a \sqrt{\,} sign is actually a surd. \sqrt{25} looks like one, but it equals the integer 5 and carries no irrationality. \sqrt{26}, sitting right next to it, is a surd — there is no way to simplify it away. The distinction matters because JEE questions routinely hide integers behind radical signs to see if you will waste time rationalising something that was rational all along.

This page trains your eye to sort them at a glance.

What counts as a surd

A surd is an irrational root of a rational number. Put differently: \sqrt[n]{a} is a surd when a is rational, n is a positive integer, and the root does not come out to a rational number.

So the test has two parts:

  1. Is the expression a root of a rational number?
  2. Is the root itself irrational?

If both answers are yes, you have a surd. If the root simplifies to a rational number — even a whole integer — it is not a surd.

Why: "surd" is a word from the Latin surdus, meaning "silent" or "deaf" — the mathematicians who coined it were translating an Arabic term for "inexpressible," meaning a root that cannot be expressed as a fraction. An expression that simplifies to an integer is fully expressible, so by the original definition, it never qualified.

The recognition recipe

For a square root \sqrt{N} of a positive integer N:

For a cube root \sqrt[3]{N}: replace "perfect square" with "perfect cube" (N = k^3). Same logic for n-th roots: check whether N is a perfect n-th power.

The full result — that \sqrt{N} is either an integer or irrational, with no fractional possibilities in between — is a classical theorem. You can read the short proof in Most Square Roots Are Irrational.

The quiz

Drag the indicator along the row of expressions. Each one is labelled surd or not a surd — check your own answer against the correct one before sliding past.

Interactive quiz sorting seven roots into surds and non-surds A horizontal row of seven radical expressions: square root of twenty-five, square root of twenty-six, square root of forty-nine, cube root of twenty-seven, cube root of twenty-eight, fourth root of sixteen, and square root of two. A slider beneath the row selects one expression at a time. For each selection, the panel above shows whether the expression is a surd or not, along with the reason. √25 √26 √49 ∛27 ∛28 ⁴√16 green dot = not a surd (simplifies to integer) orange dot = surd (irrational, stays under radical) Check each expression against the dot colour before reading the verdict below.
Seven radicals, only two of which are surds. The rest all hide an integer or rational value. $\sqrt{25} = 5$, $\sqrt{49} = 7$, $\sqrt[3]{27} = 3$, and $\sqrt[4]{16} = 2$ — all clean integers, all **not** surds. Meanwhile $\sqrt{26}$ and $\sqrt[3]{28}$ are genuinely irrational, and no simplification will get rid of the radical sign.

The verdicts, one by one

Why it matters in a JEE problem

Imagine a question asks you to rationalise \dfrac{3}{\sqrt{49} + \sqrt{16}}. A second's hesitation and you might reach for the conjugate \sqrt{49} - \sqrt{16} and the difference-of-squares trick. But \sqrt{49} + \sqrt{16} = 7 + 4 = 11, a plain integer. The "fraction" is just 3/11 in disguise. Spotting the non-surds saves the work.

Why: the surd/non-surd distinction is not just terminology. It tells you whether a radical is an object you need to manipulate (a surd) or a value you can evaluate on sight (a non-surd). Treating them the same way is the slowest possible approach to a JEE paper.

A one-line checklist

Before you rationalise, simplify, or square anything:

If yes — it is not a surd, and you can evaluate it directly. If no — treat it as a surd and use the rules from Roots and Radicals.

Related: Roots and Radicals · Is Root-4 Irrational Because It's Under a Root Sign? · When You See a Root of a Non-Perfect-Square, Default to Irrational · Spot a Conjugate Surd in a JEE Problem and Rationalise on Sight · What's the Difference Between a Radical and a Surd?