Square Root Calculator

Use the first tab for degree 2, the second for degree 3, or Other for any whole degree from 2 to 100. The result panel reports the decimal answer, checks whether it is a whole number and shows its power relationship to the input. It evaluates one root, not a symbolic expression containing several radicals.

Choose the degree before entering a negative value

The square-root symbol denotes the nonnegative, principal root: √25 = 5. Solving x² = 25 is a different question and gives both 5 and -5. A cube root reverses the third power, while the general nth-root relationship is ⁿ√x = x^1/n. The cube root calculator option gives ∛125 = 5.

Even-degree roots of negative numbers are not real. Odd degrees preserve the sign, so ∛(-8) = -2 and ⁵√(-32) = -2. Switching to Other lets you test higher degrees without changing the number already entered.

Exact radicals and decimal approximations answer different questions

Radical numbers such as √2 can be kept in exact symbolic form or written as rounded decimals. The symbol √2 is exact; 1.4142 is only an approximation. A measurement may need four decimal places, while an algebra exercise usually keeps the radical.

Are Square Roots Rational?

A perfect-square integer has a rational root: √81 = 9. If a positive integer is not a perfect square, its degree-two root is irrational. Its decimal expansion neither ends nor repeats. Cube roots follow the corresponding perfect-cube test.

To estimate a square root without a calculator, start with nearby squares. Since 49 < 50 < 64, √50 must be between 7 and 8. The displayed 7.0710678119 passes that quick range check before you use it in another calculation.

Three operations require three different checks

Multiplication and division keep the same degree

For a ≥ 0 and b ≥ 0, multiplying square roots uses √a × √b = √(ab); for example, √3 × √12 = 6. Multiplying radical expressions with coefficients treats the coefficients separately, so 2√3 × 5√2 = 10√6.

Dividing square roots uses √a / √b = √(a/b), for a ≥ 0 and b > 0. Dividing radical expressions follows the same separation: 12√10 / 3√2 becomes 4√5. Roots with different degrees should first be written as fractional exponents.

Adding degree-two roots means combining coefficients of matching radicals, not adding the radicands. Simplify √8 + √2 to 2√2 + √2, then obtain 3√2. Adding radical expressions such as √2 + √3 stops there because the radicands differ.

Precision and the power statement

The decimal calculation uses JavaScript Number precision and displays up to ten decimal places, with scientific notation for very small or large values. Whole-number status is checked separately against the original input using BigInt, not inferred from a decimal that happens to round to an integer. A non-integer result can still be rational: √0.25 = 0.5.

On the English page, 1,234 means 1234 and 1.234 means a decimal. In scientific notation, both 1.234e3 and 1,234e3 mean 1234. Invalid or changed input hides the previous answer. The power statement shows the relationship to the input; it is not an independent second calculation. It uses = only when an exact integer root and the input are fully represented on screen, otherwise ≈.