The square root of a number is the value that, when multiplied by itself, gives the original number. It is one of the most common operations in math class, engineering, statistics, finance, and everyday problem solving — from finding the side of a square room to converting a variance into a standard deviation. This square root calculator returns the root instantly, and at the same time shows you the square, the matching nth root, and whether the number is a perfect square.
Type any non-negative number into the box and the result updates as you type. Switch the root degree from 2 to 3 (cube root) or 4 (fourth root) if you need a higher root instead of a plain square root.
How to Use
- Enter the Number (x) you want to take the root of. For a classic square root, this is the value under the radical sign.
- Leave Root Degree at 2 for a square root. Change it to 3 for a cube root, 4 for a fourth root, and so on.
- Read the result card: the highlighted Square Root is the main answer. Square (x squared) shows x × x as a check, and Nth Root applies the degree you chose.
- Check Perfect Square? — it reads Yes when your root is a whole integer and No otherwise, which is handy when you need an exact answer rather than a decimal.
- If you need to double-check, multiply the displayed square root by itself: you should recover the original number.
The engine uses Newton's iteration — repeatedly averaging a guess with x divided by that guess — which converges to the correctly rounded root within a handful of steps.
The Square Root Formula
By definition, y is the square root of x when y × y = x, written y = √x. For a perfect square the root is an integer: √144 = 12, because 12 × 12 = 144. For most real numbers the root is irrational, an endless decimal with no repeating pattern, which is exactly why a calculator is useful.
Newton's method starts with any guess g and replaces it with (g + x/g) / 2. Feed 144 and start from a rough guess of 10: the next estimate is (10 + 14.4)/2 = 12.2, then 12.003, and within two or three passes it lands on 12. The same routine generalizes to any nth root: the degree-3 root of 27 is 3, and the degree-4 root of 81 is also 3.
A key rule: the square root of a negative number is not a real number. If you type -16, this calculator shows an error instead of a meaningless decimal. In advanced math, the answer would be an imaginary number (4i), but that is outside the scope of this everyday tool.
When You Actually Need a Square Root
- Geometry: if a square patio covers 144 square feet, the side length is √144 = 12 feet. Knowing the side tells you how much fencing or trim to buy.
- Statistics: standard deviation is the square root of variance. A portfolio with a variance of 0.04 has an annualized volatility of √0.04 = 0.20, or 20%.
- Finance: compound growth, CAGR conversions, and price-risk calculations all invert through a square root when you move between periods.
- Physics and engineering: the distance formula, pendulum period, and many force equations embed √x directly.
- Everyday estimating: if a square carpet costs $8 per square foot and covers 144 ft², the room's width is the root of that area, not the area itself.
Examples
Example 1 — Perfect square. Enter 144 with root degree 2. The square root is 12.0, the square is 20,736, and Perfect Square reads Yes. Multiply 12 by 12 and you recover 144 exactly.
Example 2 — Irrational root. Enter 2. The square root is approximately 1.4142, the square is 4, and Perfect Square reads No. This is the famous Pythagorean diagonal of a 1×1 square; the decimal never ends and never repeats.
Example 3 — Large number. Enter 1,234,567. The root is about 1,111.11, while the square balloons to 1,524,155,677,489. The calculator handles numbers past a million without breaking, and it flags that the result is not a perfect square.
Example 4 — Cube root. Change root degree to 3 and enter 27. The nth root is exactly 3.0, while the plain square root of 27 is about 5.196. This shows why the degree control matters: 3 is the side length of a cube whose volume is 27 cubic units.
FAQ
What is a square root?
The square root of a number x is the value y such that y × y = x. For example, the square root of 144 is 12 because 12 × 12 = 144. It is written with the radical sign √x.
Is the square root of a negative number possible?
Not as a real number. There is no real number you can square to get -16, because a positive times a positive is positive and a negative times a negative is also positive. For negative inputs this calculator returns an error; in complex math the answer would be an imaginary number such as 4i.
What is a perfect square?
A perfect square is a whole number whose square root is also a whole integer. Examples include 1, 4, 9, 16, 25, 144, and 1,000,000 (root 1000). On this calculator the Perfect Square? line reads Yes for such numbers and No otherwise. Note that a decimal such as 2.25 has a clean root (1.5) but is not itself an integer, so the flag reads No.
How is a square root different from a cube root?
A square root (degree 2) asks "what times itself twice gives x?", while a cube root (degree 3) asks "what times itself three times gives x?". The cube root of 27 is 3 because 3 × 3 × 3 = 27. Use the Root Degree control to switch between them on this page.
How accurate is this calculator?
The engine uses Newton's method until the estimate stops changing within 12 decimal places, matching the correctly rounded result you would get from Python's standard math library. For everyday use the displayed two- or four-decimal result is exact well beyond what you need.
Can I take the square root of a decimal?
Yes. Enter any non-negative decimal, such as 2.25, and the root appears immediately: √2.25 = 1.5. Decimals are handled exactly like whole numbers by the same iteration.
What if I enter root degree 0 or a negative degree?
The root index must be at least 1. A degree of 0 would mean x^(1/0), which is division by zero, and a negative degree would turn the operation into a reciprocal power rather than a root — both are rejected with an error. Degree 1 is allowed but trivial: the first root of any number is the number itself. Use degree 2 for square roots, degree 3 for cube roots, and higher integers for fourth, fifth, and beyond.
Disclaimer
This square root calculator is provided for informational purposes only and does not constitute mathematical, financial, engineering, or educational advice. Results are based on the non-negative number you enter and standard Newton iteration; roots of negative numbers are rejected rather than converted to complex values. Always double-check critical calculations with a recognized reference before relying on them for decisions.