Square Root Calculator

How to Use the Square Root Calculator

A root undoes a power. The square root asks what number, multiplied by itself, gives you this one — and the same question with three copies gives a cube root, with n copies an nth root. This calculator does all of them and shows the exact radical form where one exists.

1. Enter the number you want the root of. Negative numbers are allowed for odd roots — the cube root of −8 is −2 — and refused for even ones, because no real number squared gives a negative.
2. Set the degree. Two for a square root, three for a cube root, four for a fourth root. The degree can be fractional, though that is unusual enough to be worth double-checking when you do it.
3. Read the radical form for square roots of whole numbers. It pulls the largest perfect-square factor out from under the sign, which is the form algebra and geometry actually want.
4. Check the raised-back tile. It takes the result to the power of the degree, which should return the number you started with. Any discrepancy is rounding rather than error.
5. Look at the perfect-square tile to see immediately whether the answer is a whole number or an irrational one whose decimals never settle into a pattern.

Every calculator returns the principal root — the positive one. It is worth remembering that x² = 9 has two solutions, 3 and −3, even though √9 means only 3.

Root Formulas

Roots are fractional exponents, and everything else follows from that.

Simplifying a radical means finding the largest square that divides the number and taking its root outside. It is not an approximation — 24√3 is exactly as precise as √1728 and considerably easier to work with, because the irrational part has been isolated into one small symbol.

Example

The square root of 1,728

1. Factorise: 1,728 = 2⁶ × 27 = 576 × 3, and 576 is 24 squared.
2. Pull the perfect square out: √1728 = √(576 × 3) = 24√3.
3. As a decimal: 24 × 1.7320508 = 41.5692193817.
4. Check by squaring: 41.5692193817² = 1,728.
5. Is it a perfect square? No — 41² is 1,681 and 42² is 1,764, so the root falls between two whole numbers.
6. The decimals never repeat and never end, because √3 is irrational and 24√3 therefore is too.

The same number, cubed instead

1,728 is a perfect cube: 12 × 12 × 12. So while its square root is irrational, its cube root is exactly 12. That combination is not unusual — a number can be a perfect cube without being a perfect square, and vice versa. It is also why 1,728 turns up in old measurement systems: it is the number of cubic inches in a cubic foot, twelve cubed.

Radical form in practice

Two hundred simplifies to 10√2, because 200 is 100 × 2. Two thousand and twenty-five is 45 exactly, since 45² = 2,025. And 169 is 13. Knowing which numbers are perfect squares makes mental estimation far easier: √1728 must be between 41 and 42 because those squares bracket it, and that check takes a second and catches an order-of-magnitude error immediately.

Which Roots Come Out Whole

The same number under several different roots.

Notice that the fourth root is the square root of the square root, and the sixth root is the square root of the cube root. Roots compose the way exponents multiply, because b^1/2 raised to the power 1/3 is b^1/6.

Whether a root is rational depends entirely on the factorisation. A whole number has a whole square root only when every prime in its factorisation appears an even number of times. 1,728 is 2⁶ × 3³: the six twos pair up neatly into 2³ = 8 outside, but the three threes leave one behind — which is exactly why the answer is 24√3 with a single 3 remaining under the sign.

The irrationality of √2 is one of the oldest results in mathematics and it has a short proof: if √2 were a fraction in lowest terms, both its parts would have to be even, which contradicts it being in lowest terms. The discovery is said to have distressed the Pythagoreans considerably, since it meant not every length could be written as a ratio of whole numbers — a genuinely unsettling thing to learn if your philosophy rests on the opposite.

Four Things About Roots That Catch People Out

Four things about roots that catch people out.

Roots do not distribute over addition. √(9 + 16) is 5, and √9 + √16 is 7. This is the single most common algebra error at school level and it survives well into adulthood. Multiplication and division are fine; addition and subtraction are not.

Every positive number has two square roots. Both 3 and −3 square to 9. The radical sign means the principal — positive — root by convention, which is why solving x² = 9 requires writing x = ±3 explicitly rather than relying on the symbol to remember it for you.

Even roots of negatives are not real. There is no real number whose square is −4, because any real number squared is positive or zero. Complex numbers extend the system to allow it; this calculator stays in the reals and returns an error instead.

Simplified form is exact, decimals are not. 24√3 is precisely correct forever; 41.5692193817 is correct to ten places and wrong at the eleventh. In geometry, physics and any calculation that will be squared again later, staying in radical form until the end avoids accumulating that error.

Estimating a square root by hand is a skill worth keeping. Find the two perfect squares that bracket your number, then interpolate: 1,728 sits between 41² = 1,681 and 42² = 1,764, closer to the lower one, so the answer is a little under 41.6. That takes seconds, needs no device, and catches the kind of error where a misplaced keystroke returns an answer ten times too large.

Roots turn up constantly outside pure arithmetic. The standard deviation calculator takes a square root at the last step to bring variance back into the original units, and the area calculator uses one to find the side of a square with a given area. Both are the same operation with different labels, which is the exponent calculator's point in reverse.