Maths
What is a square root?
Quick answer
The square root of a number is the value that, multiplied by itself, gives that number. The square root of 25 is 5, because 5 × 5 = 25. The sign is √.
In one sentence
A square root undoes squaring: it asks "what number times itself gives this?"
Explained
You know that 6 × 6 = 36, so 36 is a square number. Going backwards, the square root of 36 is 6. We write it √36 = 6. The tick-like sign is called a radical or root sign.
Picture a square carpet. If you know its area is 49 m², the square root tells you the length of one side: √49 = 7 m. That is where the name comes from: the root is the side of the square.
mat
For square numbers, the root is a whole number:
| Number | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 | 144 |
|---|---|---|---|---|---|---|---|---|---|---|---|
| √ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 12 |
Most numbers are not square numbers, and their roots are long decimals that never end or repeat. √2 = 1.41421…, for example. You can still estimate them using the squares you know. For √30: 5² = 25 and 6² = 36, so √30 is between 5 and 6, a little closer to 5. A calculator says 5.477…
Strictly, every positive number has two square roots, because (−6) × (−6) = 36 as well. The √ sign means the positive one. When solving x² = 36, older students write x = 6 or x = −6 (often shown as ±6).
There are other roots too. The cube root ∛ undoes cubing: ∛64 = 4 because 4 × 4 × 4 = 64.
Square roots are essential for Pythagoras' theorem, where you often finish by finding a root.
Examples
- √9 = 3
- √100 = 10
- √400 = 20 (since 20 × 20 = 400)
- √0.25 = 0.5 (since 0.5 × 0.5 = 0.25)
- √20 is about 4.47 (between 4 and 5)
- ∛125 = 5
Common mistakes
- Halving instead of rooting. √16 is 4, not 8. Fix: check by multiplying your answer by itself.
- Thinking √ of a big number is always a decimal. √10,000 = 100 exactly.
- Adding roots like ordinary numbers. √9 + √16 = 3 + 4 = 7, but √25 = 5. You can't just add under the root sign.
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Related topics
Square numbers
A square number is what you get when you multiply a whole number by itself, like 4 × 4 = 16. The first square numbers are 1, 4, 9, 16, 25 and 36.
Powers and indices
A power (also called an index or exponent) is a small raised number showing how many times to multiply a number by itself. So 2⁴ means 2 × 2 × 2 × 2 = 16.
Pythagoras' theorem
Pythagoras' theorem says that in a right-angled triangle, a² + b² = c², where c is the longest side (the hypotenuse). For sides 3 and 4, the hypotenuse is 5.
Estimation
Estimating means making a sensible guess at an answer, usually by rounding the numbers first. It tells you roughly what to expect and helps you spot mistakes.
Cube numbers and cube roots
A cube number is a number multiplied by itself three times, like 4³ = 4 × 4 × 4 = 64. The cube root undoes it, so ∛64 = 4.
Surds (radicals)
A surd is a root that cannot be written as a whole number or fraction, like √2 or √5. In the US they are called radicals. Simplify them with √(ab) = √a × √b, so √12 = 2√3.