What does the √ symbol mean?

The √ symbol, called the radical sign, asks for the square root of the number beneath it. So √81 means the number that gives 81 when multiplied by itself, which is 9. Strictly, −9 works too, since (−9) × (−9) = 81, but √ always names the positive root. Add a small 3 to the sign, as in ∛, and it asks for a cube root instead: ∛8 = 2.

How do you find a square root without a calculator?

To find a square root without a calculator, trap it between squares you already know. For √576, start with round tens: 20² = 400 is too small and 30² = 900 is too big. Next, 25² = 625 is only a little too big, so the root sits just below 25. Try 24: 24 × 24 = 576, so √576 = 24.

How can you remember Pythagorean triples?

Remember one triple and scale it. In a right triangle with short sides 3 and 4, the long side, called the hypotenuse, is √(3² + 4²) = √25 = 5. Double every number and 6, 8, 10 still works; multiply by 4 and you get 12, 16, 20. Two more families worth knowing start at 5, 12, 13 and at 8, 15, 17.

Why does my child think the square root of 16 is 8?

Many children halve the number, partly because halving gives the right answer for √4 = 2, often the first root they meet. A square root is not a half: √16 = 4, since 4 × 4 = 16, while 8 × 8 = 64. Ask your child to multiply each answer by itself. If that does not give back the number under the sign, the root is wrong.

Is this roots game free for teachers, and what does a lesson with it look like?

A lesson with it costs nothing: every level, Olympic roots included, opens in the browser on school laptops or tablets, with no sign-ups. For square root practice at the board, write 150 and ask which two whole numbers its root lies between. Pupils list squares until they pass it: 12² = 144 and 13² = 169, so √150 is between 12 and 13. The schools plan, priced per student, then shows on a dashboard who is getting quicker at roots.

Math practice · reimagined

Square Roots Game

This square root game starts with perfect squares up to 400, like √289, and climbs through cube roots to Olympic roots worked to two decimal places, such as √50 ≈ 7.07. Each root you recognise at a glance becomes a blow your rival never gets to throw.

Ages 4–99+ · no download · plays in your browser · 23 languages

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What is a square root?

A square root of a number is the value that makes that number when multiplied by itself, and its sign is √. The name comes from shapes. A chessboard is a square of 64 small squares, with 8 along each side, so √64 = 8. Numbers like 1, 4, 9 and 64 are called perfect squares, because their roots are whole numbers.

Every level of Square root

Example: √100 − √169

Work out the first root on its own: 10 × 10 = 100, so √100 = 10.
Then the second root: 13 × 13 = 169, so √169 = 13.
Now subtract the two roots, not the numbers under the signs: 10 − 13 = −3.
The result is below zero, so enter 3 and tap the − key to make it −3.

Learn it step by step

A square root runs squaring backwards. It asks: "what number times itself gives this?"

√49 = 7 because 7 × 7 = 49. √144 = 12 because 12 × 12 = 144. If you know your squares, you know your roots — they're the same facts read the other way.

These all come out to whole numbers. Type the root.

Tips and common mistakes

Perfect squares never end in 2, 3, 7 or 8. If one ends in 6, its root ends in 4 or 6, as in √196 = 14.
Roots do not add like ordinary numbers: √9 + √16 is 3 + 4 = 7, while √(9 + 16) is √25 = 5. Check whether the plus sits inside the root or outside it.
Give your child 25 coins to lay out as a full square. Each side holds 5, so √25 = 5. Then try 30 coins, which no square can hold.

How to play

A root appears for both fighters, from √289 to ∛343 or √(15² + 8²). Type the answer on the keypad: a whole number on most levels, with the − key when √4 − √49 drops below zero, and every decimal place on the Olympic ones. Answer first and Onion strikes; a root typed in a flash hits harder.

— Quick answers —

Square root: the short FAQ

What does the √ symbol mean?: The √ symbol, called the radical sign, asks for the square root of the number beneath it. So √81 means the number that gives 81 when multiplied by itself, which is 9. Strictly, −9 works too, since (−9) × (−9) = 81, but √ always names the positive root. Add a small 3 to the sign, as in ∛, and it asks for a cube root instead: ∛8 = 2.
How do you find a square root without a calculator?: To find a square root without a calculator, trap it between squares you already know. For √576, start with round tens: 20² = 400 is too small and 30² = 900 is too big. Next, 25² = 625 is only a little too big, so the root sits just below 25. Try 24: 24 × 24 = 576, so √576 = 24.
How can you remember Pythagorean triples?: Remember one triple and scale it. In a right triangle with short sides 3 and 4, the long side, called the hypotenuse, is √(3² + 4²) = √25 = 5. Double every number and 6, 8, 10 still works; multiply by 4 and you get 12, 16, 20. Two more families worth knowing start at 5, 12, 13 and at 8, 15, 17.
Why does my child think the square root of 16 is 8?: Many children halve the number, partly because halving gives the right answer for √4 = 2, often the first root they meet. A square root is not a half: √16 = 4, since 4 × 4 = 16, while 8 × 8 = 64. Ask your child to multiply each answer by itself. If that does not give back the number under the sign, the root is wrong.
Is this roots game free for teachers, and what does a lesson with it look like?: A lesson with it costs nothing: every level, Olympic roots included, opens in the browser on school laptops or tablets, with no sign-ups. For square root practice at the board, write 150 and ask which two whole numbers its root lies between. Pupils list squares until they pass it: 12² = 144 and 13² = 169, so √150 is between 12 and 13. The schools plan, priced per student, then shows on a dashboard who is getting quicker at roots.

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