Root Calculator
Find the square root, cube root or any nth root of a number, with the simplest radical form and every step shown.
Solving r × r = x for r.
7.071067812 × 7.071067812 ≈ 50.
50 lies between the perfect squares 49 and 64.
Simplest radical form
Factors that appear as a full square move outside the radical sign.
Check: 5 × √2 = 5 × 1.414213562 = 7.071067812.
Your calculation
The same steps the calculator runs, with your numbers.
Newton's method from a starting guess
Each row improves the guess. Two or three rounds usually fix seven or more digits.
| Step | Guess | Guess² |
|---|
Perfect squares around your number
Your number is highlighted between its nearest perfect powers.
| Number | Square root |
|---|
Results use double precision arithmetic and are rounded to 10 significant digits. Only real roots are shown.
What Is a Root?
A root undoes a power. The square root of x is the number that gives x when used as a factor twice, so √49 = 7 because 7 × 7 = 49. The cube root is used as a factor three times, so ∛125 = 5 because 5 × 5 × 5 = 125. In general, the nth root of x, written ⁿ√x, is the number r with rⁿ = x.
The parts of a root have names. The symbol √ is the radical sign, the number under it is the radicand, and the small number n is the index. A square root has index 2, which is not written.
Every positive number has two square roots, one positive and one negative: 7 and −7 both square to 49. The radical sign always means the principal root, the one that is 0 or positive, so √49 = 7. Solving x² = 49 still gives both answers, x = ±7.
How to Use the Root Calculator
- Choose the root. Square √ for index 2, Cube ∛ for index 3, or Nth root for any whole number index from 2 to 99.
- Enter the number x. Whole numbers, decimals and negative numbers are accepted. Negative numbers have real roots only for odd indexes such as 3 and 5.
- Read the result. The large number is the decimal root to 10 significant digits. Below it you get the simplest radical form, the same number under other roots, the working step by step, Newton's method iterations and a table of nearby perfect powers.
Results update as you type. Press Enter or Calculate on a phone to jump to the answer. The quick examples load common homework cases, including a perfect square, a decimal with an exact root and a negative cube root.
Root Formulas
The product and quotient rules hold for roots of positive numbers and are the basis for simplifying radicals. There is no matching rule for sums: √(9 + 16) = √25 = 5, not 3 + 4 = 7.
How to Simplify a Radical
- Write the radicand as a product of primes.
- Group equal primes in sets of n, the index.
- Move one factor from each full set outside the radical. Leave the rest inside.
A radical is in simplest form when the radicand has no factor that is a perfect nth power other than 1. √30 is already simple because 30 = 2 × 3 × 5 has no repeated prime.
How to Calculate a Square Root by Hand
The Babylonian method is Newton's method applied to r² = x. It starts from a guess and averages the guess with x divided by the guess. The steps below find √50.
- Bracket the root. 7² = 49 and 8² = 64, so √50 is between 7 and 8 and very close to 7. Start with r₀ = 7.
- First round. r₁ = (7 + 50 ÷ 7) ÷ 2 = (7 + 7.142857143) ÷ 2 = 7.071428571. Check: 7.071428571² ≈ 50.005102.
- Second round. r₂ = (7.071428571 + 50 ÷ 7.071428571) ÷ 2 = (7.071428571 + 7.070707071) ÷ 2 = 7.071067821.
- Compare. The true value is 7.071067812, so two rounds already give seven correct decimal places.
Newton's method roughly doubles the number of correct digits each round once the guess is close. A good first guess from the nearest perfect power saves rounds, which is why the calculator starts from the whole number part of the root.
Perfect Squares, Cubes and Roots Table
Roots of whole numbers that are not perfect powers are irrational, so the decimals below are rounded to four places.
| n | n² | n³ | √n | ∛n |
|---|---|---|---|---|
| 1 | 1 | 1 | 1 | 1 |
| 2 | 4 | 8 | 1.4142 | 1.2599 |
| 3 | 9 | 27 | 1.7321 | 1.4422 |
| 4 | 16 | 64 | 2 | 1.5874 |
| 5 | 25 | 125 | 2.2361 | 1.7100 |
| 6 | 36 | 216 | 2.4495 | 1.8171 |
| 7 | 49 | 343 | 2.6458 | 1.9129 |
| 8 | 64 | 512 | 2.8284 | 2 |
| 9 | 81 | 729 | 3 | 2.0801 |
| 10 | 100 | 1,000 | 3.1623 | 2.1544 |
| 11 | 121 | 1,331 | 3.3166 | 2.2240 |
| 12 | 144 | 1,728 | 3.4641 | 2.2894 |
Read the table both ways. The n² column lists perfect squares, so √144 = 12. The √n column gives the root of small numbers, so √7 ≈ 2.6458.
Roots of Negative Numbers
Odd roots of negative numbers are real and negative: ∛−27 = −3 because (−3) × (−3) × (−3) = −27, and the fifth root of −32 is −2.
Even roots of negative numbers are not real, because any real number raised to an even power is 0 or positive. In the complex numbers, √−16 = 4i, where i² = −1. The calculator shows a message instead of a misleading value when an even root has no real answer.
The root of 0 is 0 for every index, and the root of 1 is 1.
Understanding Your Result
If the radicand is a perfect power, such as 144 for square roots or 1,000 for cube roots, the root is a whole number. For any other whole number the root is irrational: its decimals never end or repeat, so the exact answer is the radical form, such as 5√2.
The decimal is rounded to 10 significant digits. Round further only at the last step of a longer problem. Rounding √2 to 1.41 and squaring gives 1.9881, not 2.
For numbers between 0 and 1 the root is larger than the number: √0.25 = 0.5.
Where Roots Are Used
Distance and the Pythagorean theorem
The long side of a right triangle is c = √(a² + b²). Legs of 3 and 4 give √25 = 5. The diagonal of a 10 m square is √200 = 10√2 ≈ 14.14 m.
Area and volume
A square room of 144 square feet has sides of √144 = 12 feet. A cube holding 1,000 cm³ has edges of ∛1,000 = 10 cm.
Statistics
The standard deviation is the square root of the variance, so a variance of 16 gives a standard deviation of 4 in the original units.
Growth rates
The average yearly growth over n years is ⁿ√(end ÷ start) − 1. An amount that doubles in 5 years grew at 21/5 − 1 ≈ 0.148698, or about 14.87% per year.
Common Mistakes
- Splitting a sum. √(9 + 16) = 5, while √9 + √16 = 7. Roots split over products and quotients only.
- Forgetting the negative solution. √49 = 7, but x² = 49 has two solutions, 7 and −7.
- Writing √(x²) = x. The correct rule is √(x²) = |x|, so √((−5)²) = 5.
- Confusing a cube root with dividing by 3. ∛27 = 3, but 27 ÷ 3 = 9.
- Taking an even root of a negative number. √−16 has no real value. Check the sign before you start.
- Leaving a perfect square inside. √72 = 3√8 is not finished, because 8 = 4 × 2 still holds a square. The simplest form is 6√2.
Related Calculations
Roots and powers are inverse operations, so the Exponent Calculator checks any root: raise the result to the index and you get the radicand back. The Log Calculator gives a third route, because ⁿ√x = 10log x ÷ n.
Square roots appear inside other formulas. The Pythagorean Theorem Calculator finds a missing side, the Quadratic Formula Calculator takes the square root of the discriminant, and the Statistics Calculator takes the square root of the variance to get the standard deviation.
Frequently Asked Questions
How do you find the square root of a number without a calculator?
Start with the nearest perfect squares, then refine with the Babylonian method, which is Newton's method for square roots. For √50, 7² = 49 and 8² = 64, so the root is just above 7. Average the guess with the number divided by the guess: (7 + 50 ÷ 7) ÷ 2 = 7.0714286. Repeat once more to get 7.0710678, which already matches the true value 7.0710678119 to seven decimal places. Each round roughly doubles the number of correct digits, so two or three rounds are enough for most homework and measurement work.
How do you simplify a square root?
Split the number under the radical into a perfect square times a leftover factor, then take the square root of the perfect square. For √50, write 50 = 25 × 2, so √50 = √25 × √2 = 5√2. For √72, the largest square factor is 36, so √72 = 6√2. A prime factorization makes this systematic: every pair of equal prime factors moves outside the radical as one factor. A square root is in simplest form when the number left inside has no square factor other than 1, as in √30, because 30 = 2 × 3 × 5.
Can you take the square root of a negative number?
Not within the real numbers. Any real number multiplied by itself gives 0 or a positive result, so no real number squared equals −16. In the complex numbers, the principal square root of −16 is 4i, where i is the imaginary unit with i² = −1, because (4i)² = 16 × i² = −16. Odd roots behave differently: the cube root of −27 is −3, because (−3) × (−3) × (−3) = −27. This calculator returns real roots of negative numbers for odd indexes and explains when an even root has no real value.
What is the difference between a square root and a cube root?
A square root of x is a number that gives x when used as a factor twice, written √x, so √64 = 8 because 8 × 8 = 64. A cube root is a number used as a factor three times, written ∛x, so ∛64 = 4 because 4 × 4 × 4 = 64. Square roots answer area questions, such as the side of a 144 square foot room being 12 feet. Cube roots answer volume questions, such as a 1,000 cm³ cube having 10 cm edges. Every real number has one real cube root, but only numbers of 0 or more have real square roots.
How do you calculate the nth root of a number?
The nth root of x is the number that gives x when used as a factor n times, and it equals x raised to the power 1/n. On a calculator, enter x, press the power key and type (1 ÷ n). For example, the fourth root of 81 is 81^0.25 = 3, because 3⁴ = 81, and the fifth root of 32 is 2, because 2⁵ = 32. Logarithms give the same answer: ⁿ√x = 10^(log₁₀ x ÷ n). Finance uses nth roots for growth rates: an investment that doubles in 5 years grew at 2^(1/5) − 1 ≈ 14.87 percent per year.
Sources & References
- OpenStax. College Algebra 2e, 1.3 Radicals and Rational Exponents (principal root, product and quotient rules, xm/n). openstax.org
- OpenStax. Elementary Algebra 2e, 9.2 Simplify Square Roots (product property and simplest form). openstax.org
- OpenStax. Calculus Volume 1, 4.9 Newton's Method (iteration formula and convergence). openstax.org
- NIST Digital Library of Mathematical Functions. §4.2 Definitions: Logarithm, Exponential, Powers (principal value of powers). dlmf.nist.gov/4.2
Results are for educational purposes. Every worked number on this page was recomputed with the same functions the calculator uses. Methodology reviewed: 2026.