Mathematics from zero
Square roots
A square room has an area of 9 square metres. How long is each wall? You need a number that, multiplied by itself, gives 9 — and that number is 3. Working backwards from 9 to 3 is taking a square root.
After this lesson you can say what a square root is, see why it undoes squaring, find the root of a perfect square, recognise which numbers are perfect squares, and estimate the root of a number that is not one.
Squaring a number multiplies it by itself. You met this as the exponent 2: 3² is
3 × 3 = 9, 5² is 5 × 5 = 25. Squaring is a forward question — start with a
number, get the result of multiplying it by itself.
The square root asks that question backwards. Instead of “what is 3 squared?”, the
square root asks “what number, squared, gives 9?” The answer is 3. We write it
with the root sign: √9 = 3, because 3² = 9. The square root undoes squaring, the
way subtraction undoes addition.
Numbers with a whole-number root are called perfect squares. The curve above plots
each number squared: 0, 1, 4, 9, 16, 25, 36. Those output values — 1, 4, 9, 16, 25, 36 and so on — are the perfect squares. Their square roots are exactly whole
numbers: √16 = 4, √25 = 5. Knowing the first few perfect squares makes their roots
instant.
Most numbers are not perfect squares — their roots fall between whole numbers.
Take √10. There is no whole number that squares to 10: 3² = 9 is too small,
4² = 16 is too big. So √10 lies between 3 and 4 — a little above 3, since 10 is
just above 9. You estimate such a root by trapping it between the two nearest perfect
squares.
Find √49.
The square root asks: what number, multiplied by itself, gives 49?
Try a few. 6 × 6 = 36 — too small. 8 × 8 = 64 — too big. 7 × 7 = 49 — exactly.
So √49 = 7. And 49 is a perfect square, because its root came out as a whole number.
Check by squaring back: 7² = 49. The root and the square are a matched pair —
each undoes the other.
Why this works
Why does the square root “undo” squaring? Because they are the same fact read in two
directions. 7² = 49 and √49 = 7 describe one relationship: 7 and 49 are partners,
linked by multiplying-by-itself. Squaring travels from the side to the area; the square
root travels back from the area to the side. Same pairing, opposite directions.
Common mistake
A common mistake is thinking √16 means 16 ÷ 2 = 8. It does not. The square root is
not halving. √16 asks which number times itself is 16, and 4 × 4 = 16, so
√16 = 4. Halving 16 gives 8, and 8 × 8 = 64, not 16. Always test a root by
squaring it back.
Find √25. Type the value.
Find √1. Type the value.
Find √100. Type the value.
Work out 6² (6 squared). Type the value.
Find √81. Type the value.
What does √36 ask you to find?
Squaring a number multiplies it by itself; the square root asks that question backwards — what number, squared, gives this? — and so it undoes squaring. Numbers whose root is a whole number are perfect squares: 1, 4, 9, 16, 25, 36. Most numbers are not perfect squares, and their roots fall between two whole numbers; you estimate such a root by trapping it between the nearest perfect squares.