Square root calculator
Enter a number, get its root or power instantly.
How do I find the square root of a number?
Find the number that times itself gives yours: √144 = 12, because 12 × 12 = 144. For numbers that are not perfect squares the root is a decimal: √2 = 1.41421…
How do I simplify a square root?
Pull out the largest perfect square factor. 72 = 36 × 2, so √72 = √36 × √2 = 6√2. Likewise √50 = 5√2. A number with no square factor, like 13, stays as √13.
What is a cube root?
The number that times itself three times gives yours: ∛125 = 5, because 5 × 5 × 5 = 125. Unlike square roots, cube roots of negative numbers are real: ∛−8 = −2.
Can I take the square root of a negative number?
Not with real numbers — no real number times itself is negative. The answer is imaginary: √−4 = 2i. This calculator works with real numbers, so it says so instead.
How do exponents work?
An exponent says how many times to multiply a number by itself: 2⁵ = 2 × 2 × 2 × 2 × 2 = 32. A fractional exponent is a root: 8^(1/3) = 2. A negative exponent is one over the power: 2⁻³ = 1/8.
How can I estimate a square root without a calculator?
Find the two perfect squares around the number. √10 lies between √9 = 3 and √16 = 4, and 10 is only 1 above 9, so a good first guess is a little over 3. The exact value is 3.1623. Then refine it with the averaging rule below, which is the method many calculators use.
Is √(a + b) the same as √a + √b?
No. √9 + √16 = 3 + 4 = 7, but √(9 + 16) = √25 = 5. Roots can be split over multiplication and division, √(ab) = √a × √b, but not over addition or subtraction.
What the calculator computes
For a square or cube root it finds the number that, multiplied by itself two or three times, gives your input; for an nth root it uses the same idea with n factors. An even root of a negative number is not a real number, so the tool reports "Not real" instead of an answer; an odd root of a negative number is fine, and ∛−27 = -3. Exact integer roots are returned as integers (the root of 125 for n = 3 is 5, not 4.999…). In power mode it computes xⁿ directly. Results are shown to ten decimal places, and the line under the result checks the answer by raising it back to the power.
Worked example: simplifying √200 and ∛54
200 = 100 × 2, and 100 is the largest perfect square dividing 200, so √200 = √100 × √2 = 10√2. As a decimal this is 14.142136, which you can check: 10 × 1.414214 ≈ 14.14214. For the cube root of 54, the largest cube factor is 27, so ∛54 = ∛27 × ∛2 = 3∛2, which is 3.779763. The calculator's "Simplified" line does the first step automatically for whole numbers; for cube roots, factor out perfect cubes (8, 27, 64, 125) yourself.
How a square root is found by hand
A classical method is to average a guess with the number divided by the guess. For √2, start with 1. Then (1 + 2/1) ÷ 2 = 1.5000, the next step gives 1.416667, then 1.41421569, and then 1.4142135624. Each step roughly doubles the number of correct digits, which is why a handful of steps is enough. The method works for any positive number and any positive starting guess.
Common mistakes
Expecting √(x²) to equal x: it equals |x|, the absolute value, so √((−5)²) = 5. Another is forgetting that every positive number has two square roots, +r and −r, although the √ sign means the positive one. With powers, the sign matters: (−4)² is 16 but −4² means −(4²) = −16, so enter the base as −4 only when you mean the whole number is squared. In power mode, a negative base with a fractional exponent such as (−8)^(1/3) has no real value in this tool; use the cube root instead, which gives −2. And a calculator shows a decimal approximation, so squaring the displayed value may differ from the original in the last digits.
Sources: OpenStax Elementary Algebra 2e, chapter 9 (roots and radicals: simplifying and the product rule for roots), and OpenStax Intermediate Algebra 2e on rational exponents; the averaging iteration is Newton's method applied to x² = a (Heron's method), described in OpenStax Calculus Volume 1, section 4.9. Related: exponent calculator, Pythagorean theorem calculator and quadratic formula calculator.