Square Root Calculator
The square root calculator finds the square root, cube root, or any nth root of a number instantly. Beyond just giving you a decimal answer, it shows whether the result is a perfect root, simplifies the radical form, and explains the long-division method manually. Whether you are solving geometry problems, working with quadratic equations, or checking financial calculations, this tool makes root calculations transparent.
Number Line Visual
How to Use
- Choose the root type from the tabs. Use square root for values like √144, cube root for values like ∛27, and nth root when you want a custom degree.
- Enter the number under the radical sign in the Number field. Negative numbers are allowed for cube roots and odd nth roots, but not for even real roots.
- If using nth root mode, enter the degree n. A fourth root asks what number multiplied by itself four times equals the input.
- Click Calculate Root. The result cards show the decimal result, whether it is a perfect root, a simplified radical when applicable, and a power-form check.
- Use the number line to compare the original input with the root. Roots are smaller than the original number for values greater than one.
Root Formulas
∛x = x^(1/3)
ⁿ√x = x^(1/n)
A square root answers the question: what number multiplied by itself gives x That is why √x is the same as x raised to the power one half. A cube root asks what number multiplied by itself three times gives x, so ∛x equals x raised to one third. An nth root generalizes the idea: n copies of the answer multiplied together produce the original number. The Babylonian method estimates square roots by repeatedly averaging a guess with x divided by that guess. If g is a guess for √x, the next guess is (g + x/g) / 2. Repeating this quickly approaches the true square root. For simplifying radicals, factor out perfect powers. For example, √8 becomes √(4 × 2), which equals 2√2 because 4 is a perfect square.
Worked Example
Find √144. Prime factorize 144: 144 = 2 × 72 = 2 × 2 × 36 = 2 × 2 × 2 × 18 = 2 × 2 × 2 × 2 × 9 = 2 × 2 × 2 × 2 × 3 × 3. Group the equal factors into pairs: (2 × 2), (2 × 2), and (3 × 3). One number from each pair comes outside the square root, giving 2 × 2 × 3 = 12. Therefore √144 = 12. Check the answer by squaring it: 12 × 12 = 144. Since the result is a whole number, 144 is a perfect square.
Perfect Squares Reference
| Number | Square | Square Root |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 2 |
| 3 | 9 | 3 |
| 4 | 16 | 4 |
| 5 | 25 | 5 |
| 6 | 36 | 6 |
| 7 | 49 | 7 |
| 8 | 64 | 8 |
| 9 | 81 | 9 |
| 10 | 100 | 10 |
| 12 | 144 | 12 |
| 15 | 225 | 15 |
| 20 | 400 | 20 |
| 25 | 625 | 25 |
| 30 | 900 | 30 |
Square Root FAQ
What is a square root
A square root of a number is a value that produces the original number when multiplied by itself. For example, 8 is the square root of 64 because 8 × 8 = 64. Every positive number has two square roots in algebra: one positive and one negative. However, the square root symbol √ usually means the principal, or non-negative, square root. Square roots are used in geometry, algebra, physics, statistics, and finance whenever a squared quantity needs to be reversed.
What are perfect squares
Perfect squares are numbers whose square roots are whole numbers. Examples include 1, 4, 9, 16, 25, 36, 49, and 64. They are called perfect because they can be arranged into a square grid with equal rows and columns. For example, 36 dots can form a 6 by 6 square, so 36 is a perfect square. Recognizing perfect squares makes mental math easier and helps simplify radicals such as √72, where a perfect square factor can be taken outside the radical.
How can I calculate square roots manually
One practical manual method is the Babylonian method. Start with a reasonable guess, divide the original number by that guess, then average the quotient and the guess. Repeat the process until the value stops changing much. For √50, guess 7. Divide 50 by 7 to get about 7.1429. Average 7 and 7.1429 to get 7.0714. Repeating gives 7.0711, which is very close to the true value. For perfect squares, prime factorization is often faster because pairs of equal factors come outside the radical.
What is an irrational number
An irrational number cannot be written exactly as a ratio of two integers. Its decimal expansion goes on forever without repeating a fixed pattern. Many square roots are irrational, such as √2, √3, √5, and √10. They are not whole numbers and cannot be expressed as exact fractions. Calculators show decimal approximations, but the exact value is still the radical form. This is why √8 is often written as 2√2 instead of 2.828427; the radical form preserves exactness while the decimal is rounded.
What is the difference between square root and cube root
A square root reverses squaring, while a cube root reverses cubing. If x² = 49, then x = 7 is the principal square root. If x³ = 27, then x = 3 is the cube root. Square roots involve two equal factors, while cube roots involve three equal factors. Negative numbers do not have real even roots, so √-9 is not real. But negative numbers do have real cube roots: ∛-8 = -2 because -2 × -2 × -2 equals -8.