Square Root Calculator
Find and simplify square roots, calculate any nth root, combine radicals, and learn the manual estimation method most calculators skip straight past.
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Most square root calculators simplify a radical, show a decimal, and stop there. This one does that, plus a mode most competitors skip entirely: a step-by-step walkthrough of how to estimate a square root by hand, using the nearest perfect squares and then refining that guess with the Babylonian method, the same iterative approach used before calculators existed.
Related Calculators
How Each Mode Works
1. Square Root and Simplify
If a number is a perfect square, its square root is a whole number. If not, the calculator factors out the largest perfect square it can find, so the answer is expressed with the smallest possible number left under the radical.
2. Nth Root
A square root is really just a root with degree 2. Cube roots (degree 3), fourth roots, and beyond follow the same underlying idea: what number, raised to that power, gives the original value back.
Cube root of 64: 4 × 4 × 4 = 64, so ∛64 = 4.
3. Adding and Subtracting Radicals
Radicals can only be combined directly if the number under the root is the same on both sides, the same rule as combining like terms in algebra. √2 + √3 cannot be simplified into a single radical, but 3√2 + 5√2 can, because both terms share the same radicand.
3√2 + 5√2 = 8√2.
4. Estimating a Square Root by Hand
Before reaching for a calculator, the first useful step is finding the two nearest perfect squares. The square root has to fall somewhere between their roots. From there, the Babylonian method (also called Heron's method) refines the estimate by repeatedly averaging a guess with the original number divided by that guess, getting closer to the true value with each pass.
Where Square Roots Actually Come Up
Geometry
The Pythagorean theorem is probably the most common place square roots appear outside a maths textbook: finding the diagonal of a rectangle, the length of a ladder needed to reach a certain height, or the distance between two points on a grid all involve taking a square root as the final step. A television listed as "55 inches" refers to the diagonal measurement, which is itself the square root of the sum of the screen's width and height squared, so even a spec sheet is quietly using this formula.
Physics and Engineering
Formulas for free-fall time, wave speed, and standard deviation in statistics all involve a square root somewhere in the calculation. The time it takes an object to fall a given height, for instance, is proportional to the square root of that height, which is why doubling a drop's height doesn't double the fall time. Engineers estimating a rough figure before running exact numbers often use the nearest-perfect-square method covered above as a quick sanity check before trusting a more precise calculation.
Algebra Coursework
Simplifying radicals and combining like terms with square roots is a recurring skill in algebra, showing up again later in the quadratic formula, which produces a square root as part of its solution. Getting comfortable with radical simplification early makes that later step much less intimidating.
Common Square Roots at a Glance
| Number | Square Root | Perfect Square? |
|---|---|---|
| 16 | 4 | Yes |
| 25 | 5 | Yes |
| 50 | 5√2 ≈ 7.071 | No |
| 72 | 6√2 ≈ 8.485 | No |
| 100 | 10 | Yes |
Common Mistakes
Frequently Asked Questions
How do I simplify a square root that isn't a perfect square?
Factor the number to find the largest perfect square that divides into it evenly, then move that factor's root outside the radical. The "Square Root & Simplify" mode above does this automatically and shows the factoring step.
Can I estimate a square root without a calculator?
Yes. Find the two nearest perfect squares to get a rough range, then refine with the Babylonian method, which repeatedly averages a guess with the number divided by that guess. The "Estimate by Hand" mode above walks through both steps.
What's the difference between a square root and a cube root?
A square root asks what number squared gives the original value. A cube root asks what number cubed (multiplied by itself three times) gives the original value. Both are handled by the "Nth Root" mode above by changing the root degree.
Why can't I add √2 and √3 into a single term?
Radicals can only combine directly when the number under the root matches on both terms, similar to how you can't combine "2 apples" and "3 oranges" into a single unit without converting them to a common category first.
What does it mean if a number under a square root is negative?
No real number squares to a negative value, so the square root of a negative number is imaginary, written using i (where i² = −1). This typically comes up in more advanced algebra rather than everyday calculations.