Practice maths class test — Year 2: radicals and rationalising.

15 questions drawn at random from a bank of 25, on the block of the programme your teacher is finishing: every attempt is a different test. No penalty for wrong answers, so answer even if you are not sure. At the end you receive by email the correct answer to every question and your weak topics — with an account you see them straight away here on the page. The step-by-step solution unlocks with your first lesson.

A class test is the written test an Italian high-school class sits after each block of the programme; this practice version is multiple choice.

Test in ItalianThe questions of this simulator are in Italian, as in the real exam.

Test prepared by Gaetano Livornese · Aerospace engineer, Politecnico di Milano · maths and physics tutor · updated 13 October 2026

The test on radicals comes in November-December of Year 2, just before the quadratic equations that need it as a prerequisite: if the chapter on radicals is not solid, the mistakes move forward and you pay for them twice. The test alternates domain restrictions, the invariance property, operations and at least two rationalisations.

The first mistake is about rationalising itself: when the denominator is a binomial with a root, for example 1 / (2 + √3), multiplying numerator and denominator only by √3 is not enough — you need the conjugate, that is 2 − √3, because only (2 + √3)(2 − √3) = 4 − 3 = 1 removes the root, using the special product (a + b)(a − b) = a² − b². Anyone who multiplies by the root alone still has a radical in the denominator and thinks they got the rest of the exercise wrong.

The second is about simplifying: the fourth root of x² simplifies to √|x|, not √x. The reason is that x² is always ≥ 0, so the root exists for every real x; but √x on its own only exists for x ≥ 0 — changing the domain of the expression while simplifying it is the mistake, not the mechanics of the calculation.

The third is about operations: √2 and the cube root of 2 are not like radicals even though the radicand is the same — to treat them together they must be brought to the same index (the LCM of 2 and 3, that is 6), not added or compared as if the index did not matter.

To prepare, do three rationalisations with a binomial in the denominator before opening the simulator, and every time you simplify a root with an even index ask yourself whether the domain has stayed the same.

What a maths test on radicals and rationalising looks like

Domain restrictions, simplifications with the invariance property, sums of like radicals, moving factors in and out of the radical sign and at least two rationalisations, one of them with a binomial in the denominator.

The mistake that costs the grade

Treating the root as if it were linear and writing √(a + b) = √a + √b. Not long after comes moving a factor under the root without checking its sign: if it is negative it has to be left outside, otherwise the value of the expression changes.

How to use this practice test

Take it once cold, without revising and with the timer on: it is there to show you where you stand, not to look good. Then look at your weak topics, revise only those — usually two or three, not the whole chapter — and retake it after a few days. The 15 questions are drawn from a bank of 25, so the second attempt is not the same test and the scores are genuinely comparable.