Squares & Square Roots Worksheets
Squares, roots of perfect squares, simplifying radicals · Grade 6
Three kinds of practice with squares and square roots on a notebook grid. Square a number, with part of the page asking the other way round (□² = 49), which is a square root before the symbol for one arrives; read the same table under the radical sign, again from either end (√144 = □ and √□ = 12); or pull the largest square factor out of a radicand that is not a square (√18 = □√□). The radical is drawn with its bar over the radicand, so a child learns to read where the root ends. Asking each fact from both ends is also what keeps a page from repeating itself: there are only eleven squares under 12 × 12. Squares run to 12 × 12 or 20 × 20, and the simplifying practice leans on the factor work the primes sheet builds: √72 is 6√2 only if you see 36 in it rather than stopping at 4.
Skills: squaring numbers, square roots, perfect squares, simplifying radicals, largest square factor. Format: printable, randomized on every load.
Sheets are randomized, free to print, and carry no answer key on purpose — why Super Awesome Math works this way.
Frequently asked questions
- Is this not too early for square roots?
- It depends where you are. Many curricula meet the radical sign a year or two after grade 6, but squaring is grade 6 work almost everywhere, and the roots practice is the same fact table read backwards rather than anything new. Start with squares; move to roots when the table is quick, and to simplifying only once prime factors are comfortable.
- Why does part of the page ask □² = 49 instead of 7² = □?
- Because that is the question a square root actually asks, and a child can answer it long before meeting the symbol. Writing it both ways in one page stops squaring from becoming a one-directional habit, so when √49 appears in the next practice it is a notation to learn rather than an idea to learn. It has a practical use too: there are only eleven squares under 12 × 12, and asking each one from both ends is what lets a page of two dozen problems avoid repeating itself.
- How do you simplify a square root like √72?
- Find the largest square that divides it. 72 is 36 × 2, so √72 is 6√2. The usual mistake is stopping at the first square you spot: 72 is also 4 × 18, which gives 2√18, and that is not finished because 18 still holds a 9. Factorizing into primes first makes it mechanical — 72 = 2 × 2 × 2 × 3 × 3, and every pair of equal primes sends one of itself outside.
- Why is the bar over the number, and does it matter?
- The bar, called a vinculum, says where the root ends. Without it √18 + 2 and √(18 + 2) look alike, and children who only ever see a bare tick mark read the next number along as being under the root too. The sheet draws it over digits and over answer boxes alike, so the habit forms from the first page.
- Which range should I start with?
- Squares to 12 × 12 covers the facts worth knowing by heart and matches the multiplication table a child already has. Twenty by twenty adds the squares from 13² to 20², which are worth recognising but rarely memorised, and it widens the simplifying practice to radicands up to 400.
Settings
- Practice: squares, square roots, or simplifying
- Range: squares to 12 × 12 or 20 × 20
- Columns: 2, 3 or 4, as many as the practice allows
- Print an answer key
Example problems
- 7² = □
- □² = 49
- √144 = □
- √□ = 12
- √18 = □√□
More Number sense
- Comparison – Greater than, less than, equal
- Rounding – Round to nearest 10, 100, 1000