Concept · Basic Math
What Is a Square Root? Perfect Squares and Surds
A square root undoes squaring. Everything awkward about them comes from the fact that squaring throws away the sign.
The inverse of squaring
The short answer to what is a square root: it is the number that, multiplied by itself, gives the number you started with. Since 7 × 7 = 49, the square root of 49 is 7.
Squaring and taking a square root are inverse operations, and every property of roots follows from that — including the two that cause trouble.
| Term | Means | In √72 |
|---|---|---|
| radical | the √ symbol itself | √ |
| radicand | the number underneath | 72 |
| surd | an irrational root left exact | 6√2 |
Every positive number has two square roots
Both 7 and −7 square to 49, because a negative times a negative is positive. So 49 has two square roots.
The symbol √ means only the positive one, called the principal root, which is why √49 is 7 and not ±7. The plus-or-minus appears when you solve an equation rather than evaluate a symbol.
- √49 = 7the symbol means the principal root only
- x² = 49an equation, not a symbol
- x = 7 or x = −7both values satisfy it
Writing only x = 7 for the equation discards a genuine solution, because squaring lost the sign on the way in.
Perfect squares, and everything else
A perfect square is a number whose root is a whole number. There are not many of them, and knowing the first dozen makes estimating far easier.
| n | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 |
|---|---|
| n² | 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 |
Every other whole number has an irrational square root — one that cannot be written as any fraction, and whose decimal never repeats or ends. √2 is the classic case, and the proof that it is irrational is one of the oldest results in mathematics.
That is why exact form matters. √50 has no decimal that is exactly right, so 5√2 is the only precise way to write it.
Estimating without a calculator
Trap the number between the two perfect squares either side of it.
- 49 < 50 < 64the perfect squares either side
- 7 < √50 < 8so the root is between 7 and 8
- ≈ 7.07and much closer to 7, since 50 is barely above 49
This is worth doing even when a calculator is available, because it catches the mistyped input. An answer of 22.4 for √50 is instantly wrong.
Where square roots come from
Square roots are not an abstraction invented for exams. They appear the moment you go from an area back to a length, and that happens constantly.
| Question | Why a root appears |
|---|---|
| A square field of 400 m² — how long is a side? | √400 = 20 m |
| A right triangle with legs 3 and 4 | the hypotenuse is √25 = 5 |
| Standard deviation from a variance | the root undoes the squaring of the differences |
| Time for a dropped object to fall | distance depends on t², so time needs a root |
The second row is the reason roots are so common in geometry. Pythagoras produces a squared length, and recovering the length itself needs the inverse operation.
Negative numbers have no real square root
No real number squared gives a negative result: a positive squared is positive, and a negative squared is positive too. So √−4 has no value among the real numbers.
It becomes possible once imaginary numbers are introduced, where i is defined as √−1 and √−4 is 2i. Until then, a negative under a square root means the question has no real answer, and saying so is correct.
Questions about what is a square root
What is a square root in simple terms?
The number that multiplied by itself gives the original. Since 7 × 7 = 49, the square root of 49 is 7.
Why does √49 equal 7 and not ±7?
Because the symbol means the principal, positive root. The ± appears when solving an equation like x² = 49, where both 7 and −7 are solutions.
Why can't you take the square root of a negative number?
No real number squared gives a negative result. It becomes possible with imaginary numbers, where i is defined as √−1.
Which numbers have whole-number square roots?
Only the perfect squares: 1, 4, 9, 16, 25, 36 and so on. Every other whole number has an irrational root.