A square root undoes squaring. The square root of 49 is 7, because 7 × 7 = 49. That is the whole definition, and everything else — perfect squares, radicals, estimation methods, the behaviour of negative numbers — follows from it. Square roots appear wherever a squared quantity must be recovered: the side of a square from its area, the hypotenuse from the other two sides, the standard deviation from the variance, the time of a pendulum swing from its length. This guide explains the concept, the methods for finding roots by hand, how to simplify radicals to their exact form, and the two facts that most often cause errors: every positive number has two square roots, and no real number has a negative square.

What a square root is

Squaring a number multiplies it by itself: 7² = 49, 1.5² = 2.25, (−4)² = 16. The square root reverses the operation: √49 = 7, √2.25 = 1.5, √16 = 4. The radical symbol √ denotes the principal root, the non-negative one. The notation is sometimes written with an index — ²√x — but the index 2 is conventionally omitted, just as the exponent 1 is omitted in algebra. A root of any other order is written with its index: ³√27 = 3, the cube root.

Two consequences follow immediately. First, squaring and square-rooting are inverse operations, but only in one direction cleanly: √(x²) = |x|, the absolute value, because squaring discards the sign first. Second, the square root of a product is the product of the roots: √(ab) = √a × √b for non-negative a and b. That identity is the engine of simplification, and it is the reason radicals can be manipulated exactly without decimals.

Perfect squares, and estimating any root

The perfect squares up to 20² are the estimation landmarks: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400. To estimate √50, bracket it: 49 < 50 < 64, so 7 < √50 < 8, and 50 is much closer to 49, so the root is just above 7 — in fact 7.071. To estimate √200: 196 < 200 < 225, so 14 < √200 < 15, just above 14 — in fact 14.142. The bracket gives the first digit; one refinement gives the second.

Refinement by trial is straightforward: square a candidate to one decimal place and adjust. For √50, 7.1² = 50.41 (high), 7.0² = 49 (low), so the root is between 7.0 and 7.1; 7.07² = 49.9849 (low by 0.015), 7.08² = 50.1264 (high), so √50 ≈ 7.07 to two decimal places. The Babylonian method converges faster: guess, divide, average, repeat — as the hub page’s companion section shows for √10, converging to 3.1623 in three rounds from a guess of 3. Either method reaches calculator accuracy in a handful of steps; the bracket is what keeps the answer sane, because it catches the order-of-magnitude errors that a mistyped calculator entry produces.

Simplifying radicals: exact answers without decimals

A radical is simplified when the number under it contains no perfect-square factor other than 1. Factor the radicand, extract the square root of the largest perfect-square factor, and leave the rest under the radical: √50 = √(25 × 2) = 5√2. √72 = √(36 × 2) = 6√2. √48 = √(16 × 3) = 4√3. √200 = √(100 × 2) = 10√2. The simplified form is exact, while 7.071, 8.485, 6.928, and 14.142 are approximations — and in algebra, the exact form is what the next step of the problem usually needs, because simplified radicals combine: 5√2 + 3√2 = 8√2, while 7.071 + 4.243 = 11.314 hides the structure.

Simplification also applies to fractions: √(1/2) = √2/2, rationalising the denominator by multiplying top and bottom by √2. And to variables: √(x²) = |x|, √(x⁴) = x², √(16x²) = 4|x|. The absolute value matters only when the variable’s sign is unknown; for a length or a distance, the positive form is the answer.

Where square roots show up: Pythagoras, distance, and area

The Pythagorean theorem is the most common source of square roots in practice: in a right triangle, the hypotenuse c satisfies c² = a² + b², so c = √(a² + b²). A 3-4-5 triangle has hypotenuse √(9 + 16) = √25 = 5. A ladder problem, a diagonal of a rectangle, and the distance between two points in a plane — distance = √((x₂ − x₁)² + (y₂ − y₁)²) — are all the same calculation. The distance formula is Pythagoras with coordinates, and it generalises to three dimensions by adding the third squared difference under the radical.

Area problems invert the relationship: the side of a square of area A is √A. A square of area 50 square metres has sides of 7.07 metres. A circle of area A has radius √(A/π). A sphere’s surface, a cube’s space diagonal (√3 times the edge), and the root-mean-square of a set of values — the square root of the mean of the squares, used in electrical engineering and statistics — are all square-root extractions of squared quantities. Whenever a formula squares something on the way in, recovering the original quantity takes a root on the way out.

Irrational roots, and the negative-number boundary

Most square roots are irrational: √2 = 1.41421356…, √3 = 1.7320508…, √5 = 2.2360679…, with decimal expansions that never terminate or repeat. Irrationality was proved for √2 in antiquity — if √2 were a ratio p/q in lowest terms, both p and q would have to be even, contradicting lowest terms — and the same argument covers every non-perfect-square integer. In practice, irrationality means every decimal root is an approximation, and the question is only how many digits the application needs: four for engineering, two for a building site, exact radical form for algebra.

Negative radicands have no real root, because every real square is non-negative. In complex numbers, √(−9) = 3i, with i² = −1, and every negative number has two complex roots, ±3i. If your work stays in real numbers — school mathematics, measurement, most applied work — the correct answer for a negative input is that no real square root exists, and a negative result under a radical in a real-world calculation usually signals an error earlier in the problem: a mis-signed term, a distance computed backwards, or a variance that came out negative. Check the setup before reaching for complex numbers.

Frequently asked questions

What is the square root of 144?

12. 12 × 12 = 144. The equation x² = 144 has two solutions, 12 and −12; the principal square root returned by a calculator is the positive one, 12.

What is the square root of 2?

Approximately 1.41421. √2 is irrational, so its decimal expansion never ends or repeats; 1.4142 is accurate to four decimal places.

How do you simplify √50?

Factor out the largest perfect square: √50 = √(25 × 2) = 5√2. The form 5√2 is exact; 7.071 is the decimal approximation.

Can a square root be negative?

The principal square root of a positive number is never negative. But the equation x² = 25 has two solutions, +5 and −5. When solving an equation by taking square roots, include both signs.

What is the square root of a negative number?

No real number. In complex arithmetic, √(−9) = 3i. In real-number work, a negative radicand means no real square root exists, and usually signals an error in the earlier steps.

How do I estimate a square root without a calculator?

Bracket it between two perfect squares, then refine by trial or by the Babylonian method (guess, divide, average, repeat). For √50: between 7 and 8, closer to 7; refining gives 7.07.