Square Root Calculator

Type in 72 and this square root calculator instantly tells you three useful things: the decimal value (√72 ≈ 8.485), that 72 is not a perfect square, and the clean simplified form √72 = 6√2. That last form is the one most teachers want on a test, and it is exact - unlike the rounded decimal, which goes on forever. Whether you need a quick answer for homework, a tidy radical for an algebra class, or the nth root for a science problem, this page does the arithmetic and shows the reasoning behind it.

What a square root actually is

The square root of a number x is the value that, when multiplied by itself, produces x. Squaring and taking a square root are inverse operations: one undoes the other. Because 6 × 6 = 36, the square root of 36 is 6. Because 1.5 × 1.5 = 2.25, the square root of 2.25 is 1.5. The radical symbol √ (called a radical sign) is the shorthand, and the number underneath it is the radicand.

The formula and definition

A square root is the special case (index 2) of the more general nth root. The defining relationships are:

Reading it in words: the square root of x is the non-negative number y whose square is x. Written with exponents, taking a square root is the same as raising to the power one-half, and the nth root is raising to the power 1/n. That exponent view is why a calculator can handle cube roots and beyond with the same machinery.

How to use this calculator

You only need one number to get started, and an optional second choice for the root:

1. Enter the number (radicand): type any value into the box - a whole number like 50, a decimal like 2.25, or even a negative number if you plan to use an odd root.
2. Pick the root (optional): leave it on "Square root" for the everyday case, or switch to cube root, 4th root, 5th root or 6th root from the selector.
3. Read the big result: the decimal value appears in the blue card. A green badge confirms when the number is a perfect square or perfect power; a gray badge tells you the root is irrational.
4. Check the simplified radical: for square roots of whole numbers, the calculator shows the exact form (for example 6√2) so you can write it on paper.
5. Verify it: the "Check the answer" card squares the result back to confirm it matches your input.

Everything updates as you type - there is no submit button to press.

Worked example 1: a perfect square

Enter 144. Because 12 × 12 = 144, the calculator returns exactly 12 and shows the green "perfect square" badge (12² = 144). There is no radical to simplify because the answer is a whole number. Perfect squares are the easy case - the decimal terminates and the result is an integer.

Worked example 2: simplifying a radical

Enter 72. The decimal value is about 8.4853, and 72 is not a perfect square. To simplify, find the largest perfect-square factor of 72: that is 36, since 72 = 36 × 2. Then √72 = √36 × √2 = 6√2. The calculator does this factoring automatically and displays 6√2. This exact form is preferred in algebra because nothing is rounded.

Worked example 3: a cube root

Switch the root selector to cube root and enter 27. The calculator returns 3, because 3 × 3 × 3 = 27, and flags it as a perfect power. Cube roots also work on negatives: ∛−8 = −2, since (−2) × (−2) × (−2) = −8. That is the key difference from square roots, which have no real result for negative inputs.

Perfect squares reference table

Memorizing the first set of perfect squares makes estimating any square root much faster. Here are the squares from 1 to 20:

To estimate √150 by hand, find the table entries it sits between: 144 (12²) and 169 (13²). So √150 is just above 12, around 12.2. The calculator confirms 12.247.

How to simplify a radical, step by step

The simplified radical form is what makes a square root "exact" without an endless decimal. The method is always the same:

1. Factor the radicand into a perfect square times whatever is left (use the largest perfect-square factor to finish in one step).
2. Split the radical: √(a × b) = √a × √b.
3. Take the root of the perfect square and leave the rest under the sign.

For √200: the largest perfect-square factor is 100, so √200 = √100 × √2 = 10√2. For √18: that is √(9 × 2) = 3√2. If the radicand has no perfect-square factor other than 1 (like √15 or √7), it is already in simplest form.

Who this calculator is for

• Students checking homework in pre-algebra, algebra, geometry and beyond, including the simplified radical form teachers ask for.
• Test-prep candidates (SAT, ACT, GRE) who need quick, accurate roots and want to verify mental estimates.
• Teachers and tutors generating worked examples or confirming an answer key on the fly.
• Engineers, scientists and analysts who need a fast nth root without opening a full math package.
• Anyone doing geometry (areas and distances), the Pythagorean theorem, or statistics (standard deviation) by hand.

Key terms explained

• Radicand: the number under the radical sign - the value you are taking the root of.
• Radical (√): the root symbol itself; "a radical" can also mean the whole expression like √2.
• Index: the small number that says which root (2 for square, 3 for cube). A bare √ means index 2.
• Principal root: the non-negative square root, which is what √ conventionally denotes.
• Perfect square: a number whose square root is a whole number (1, 4, 9, 16, …).
• Irrational number: a value whose decimal never ends or repeats, such as √2 or √3.
• Surd: the British/exam term for an irrational root left in radical form, like 6√2.

Square roots in the real world

Square roots are not just a classroom exercise - they reverse any "squaring" relationship. The Pythagorean theorem, c = √(a² + b²), uses one to find the long side of a right triangle from the other two; our Pythagorean Theorem Calculator does this automatically. In statistics, the standard deviation is the square root of the variance, turning squared differences back into the original units - the Standard Deviation Calculator handles that whole chain. In finance, volatility is scaled by the square root of time (the "square-root-of-time rule"). In geometry and construction, if you know a square's area you take the square root to recover the side length. Each is the same idea: an area or a sum of squares needs a square root to become a length again.

Estimating a square root without a calculator

Even with this tool handy, the mental method is worth knowing. Bracket the number between the two nearest perfect squares, then interpolate. For √50: it sits between 49 (7²) and 64 (8²), and 50 is just past 49, so the answer is a hair above 7 - about 7.07. For a closer guess, use the "divide and average" trick: guess 7.1, divide 50 ÷ 7.1 ≈ 7.04, average the two to get ≈ 7.07. Repeat once and you are within a thousandth. This is essentially how computers find roots too.

Square root vs. squaring: inverse operations

The fastest way to keep square roots straight is to treat them as the reverse gear of squaring. Squaring takes a number and multiplies it by itself (7 becomes 49); the square root takes you back the other way (49 becomes 7). Because they undo each other, √(x²) = x for any non-negative x, and (√x)² = x for any x ≥ 0. This is exactly why a square root "feels" like division does to multiplication: it is an inverse. The same logic scales up - cubing pairs with the cube root, and raising to the nth power pairs with the nth root. If you ever forget which direction you are going, square your answer and check: a correct √50 ≈ 7.07 must give back roughly 50 when you square it (7.07 × 7.07 ≈ 50.0). The calculator's "Check the answer" card automates exactly this sanity test, and the Exponent Calculator is the tool for the forward direction.

Square roots of common numbers

For numbers that are not perfect squares, the square root is an irrational decimal, but a handful of them come up so often that they are worth recognizing on sight. Here are the rounded values for the small whole numbers people look up most:

Notice that several of these share the same √2 or √3 "tail" - that is the simplified radical at work, and it is why 6√2 and 5√2 look related even though 72 and 50 do not. The decimals are rounded to five places; the radical forms in the third column are exact.

Notation, history and the √ symbol

The radical sign √ is one of the older pieces of math notation still in everyday use. It is generally traced to the German mathematician Christoff Rudolff, who used an early form of it in 1525; the horizontal bar (the vinculum) drawn over the radicand to show how much is "inside" the root came a little later. One common explanation is that the symbol began as a stylized lowercase "r" for the Latin radix, meaning "root" - the same word that gives us "radical" and "radish." Today the bare √ always means the square root (index 2), and a small raised number to the left, as in ∛ or ⁴√, sets a different index. Understanding that √ is shorthand for "the principal solution of y² = x" is what ties the symbol back to the formula at the top of this page.

Tips for accurate results

• Prefer the radical form for exact answers. 6√2 is exact; 8.485 is rounded. Use the decimal only when you need a number to plug in.
• Convert fractions to decimals first. To find √(9/16), note it equals 0.5625, and √0.5625 = 0.75 (which is also 3/4).
• Watch the sign. If an equation reads x² = 49, remember x can be 7 or −7; the radical √49 alone gives only the positive root, 7.
• Use odd roots for negatives. A cube root of a negative is real (∛−27 = −3), but a square root of a negative is not.

Related concepts and calculators

Square roots connect to several neighboring topics. If you are working with powers and exponents, remember a root is just a fractional exponent (√x = x^0.5); the Exponent Calculator raises numbers to any power, which is the inverse operation. For very large or very small roots, the Scientific Calculator adds trig, logs and exponent functions. When a problem mixes roots with fractions or ratios, the Fraction Calculator and Ratio Calculator help you prepare the numbers first. And the Percentage Calculator and Average Calculator round out the everyday math toolkit. Each tackles a different operation, but they share the same goal: a fast, correct answer with the working shown.

Sources

• NIST Digital Library of Mathematical Functions - elementary functions, powers and roots.
• IEEE Standard for Floating-Point Arithmetic (IEEE 754) - defines the double-precision sqrt operation this calculator relies on.
• NIST/SEMATECH e-Handbook of Statistical Methods - standard deviation as the square root of the variance.