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Rational or irrational roots worksheets

A root sign does not make a number irrational — √16 is 4. These free printable worksheets practice telling a rational root from an irrational one, perfect squares beside their neighbors.

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Rational or irrational roots
Rational or irrational roots worksheet — Perfect squares and neighbors, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

√2 is irrational; √16 is 4, which is as rational as a number gets. A child who has just met the word alongside the symbol calls both irrational, so these sheets deal perfect squares and their close neighbors together — √49 and √48 look alike on the page and have opposite answers. Working the root out first is what settles it.

Why this sheet works

This is the roots-only cut of the rational-or-irrational sort, kept apart from the fractions, decimals and π that fill the mixed page, so the perfect-square misreading gets its own practice. A child does not have to compute √48 to answer — only to notice that 48 is not a perfect square, so its root never comes out and the number is irrational. Recognizing the perfect squares up to a few hundred is the real work.

How this one works

One question from this sheet, worked through a step at a time.

∛343

∛343
∛343
it comes out exactly
∛343isrational
  1. One question decides it: can this be written as a fraction of two whole numbers? Not "does it look tidy", and not "is there a root sign" — those are the two guesses that go wrong.
  2. Work it out rather than judging by the sign: it is exactly 7, and any whole number counts as a fraction over one.
  3. So it CAN be written as a fraction. That makes it rational.

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