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Simplifying radicals worksheets

Simplifying a radical is finding the largest perfect square hidden inside it. These free printable worksheets ask for the coefficient and what stays under the sign, and the answer key shows the split that produced them.

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Simplifying radicals
Simplifying radicals worksheet — Square and cube roots, free printable with answer key

The answer key prints on a separate page.

What's on this sheet

The mistake this skill collects is stopping early. √864 = 4√216 is a true sentence and not an answer, and a student who has been told only “pull out a square factor” has done exactly what they were told. Every walk here names the property that ends the search instead: what is left has no square factor of its own.

Why this sheet works

The cube-root sheet is the same three sentences with a different power in them, which is why the two can share a page. On a mixed sheet the index is the thing to read first, and it is set as a raised digit against the radical rather than beside it, because 3√8 and ∛8 are different numbers.

How this one works

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

√375

375 = 25 × 15
√375 = √25 × √15
√375 = 5√15
  1. Split 375 into a perfect square times whatever is left: 375 = 25 × 15. 25 is 5 squared, and 15 has no square factor of its own — that last part is what makes this the LARGEST perfect square inside, and it is where the answer stops.
  2. A root splits across a product, so the one root becomes two.
  3. √25 is 5, so it comes out in front. √15 stays under the sign, because 15 has nothing more to give.

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