This root calculator lets you find square roots, cube roots, and custom n-th roots of any number instantly. It works for positive and negative numbers, shows exact values for perfect roots, and provides step-by-step solutions for learning and verification.
- Square root (√x): the number that multiplies by itself to give x.
- Cube root (∛x): the number that multiplies by itself three times to give x.
- n-th root (ⁿ√x): the number that, raised to the power n, equals x.
- Exact roots: shown when numbers are perfect powers (e.g., √49 = 7).
- Simplified radicals: square roots expressed as a√b when possible (e.g., √50 = 5√2).
How to use: In Simple mode, choose square or cube root and enter your number. In Advanced mode, select any root value (n), show ± even roots, view exact forms, or see simplified radical results. Step-by-step explanations are provided for students and teachers.
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Root Calculator
Find square roots, cube roots, and custom n-th roots. Use Advanced for ± even roots, exact forms, and detailed steps.
Understanding roots in mathematics
A root is the inverse of exponentiation. If an = x, then a is called the n-th root of x. Roots are used in algebra, geometry, physics, finance, and everyday problem solving. The most common are square roots (√x) and cube roots (∛x).
Square roots explained
The square root of x is the number that, when multiplied by itself, equals x. Example: √36 = 6 because 6 × 6 = 36. Square roots of perfect squares are whole numbers, while others are decimals or simplified radicals.
Cube roots explained
The cube root of x is the number that, when multiplied three times by itself, equals x. Example: ∛27 = 3 because 3 × 3 × 3 = 27. Unlike square roots, cube roots can be negative: ∛(−27) = −3.
n-th roots and general formula
The n-th root of x is written as ⁿ√x and defined as x1/n. Example: 4th root of 16 is 2, because 24 = 16. For even n, negative x has no real root. For odd n, negative values are allowed.
Worked examples
- √64 = 8 (perfect square)
- √50 = 5√2 (simplified radical form)
- ∛125 = 5 (perfect cube)
- ∛(−8) = −2 (negative cube root)
- ⁴√81 = 3 (4th root)
- ⁵√243 = 3 (5th root)
Quick reference table
| Number (x) | √x (Square root) | ∛x (Cube root) | ⁴√x (4th root) |
|---|---|---|---|
| 4 | 2 | 1.587 | ~1.414 |
| 9 | 3 | 2.080 | ~1.732 |
| 16 | 4 | 2.520 | 2 |
| 25 | 5 | 2.924 | ~2.236 |
| 64 | 8 | 4 | ~2.828 |
This table shows exact and approximate root values for common numbers, useful for quick checks.
Why roots matter for students
Roots are a core part of school mathematics. They appear in solving quadratic equations, calculating areas in geometry, simplifying radicals in algebra, and even in real-world applications like finance (compound interest) and physics (wave functions).
Limitations of calculators
Root calculators are designed for quick answers and learning support, but they don’t replace understanding the steps. Teachers usually require working to be shown. Use this tool to check homework, not to skip the learning process.
Root Calculator FAQs
A power raises a number to an exponent (23 = 8). A root finds the inverse (∛8 = 2).
Because both positive and negative numbers can square to the same result. Example: √25 = ±5, since (5 × 5) and (−5 × −5) both equal 25.
Yes, but only for odd roots. Example: ∛(−27) = −3. Even roots of negatives are not real numbers; they are complex numbers.
A simplified radical expresses a root in lowest form. Example: √50 = 5√2. The calculator shows this in Advanced mode.
The principal root is the positive solution of an even root. For example, √9 = 3 is called the principal square root.
The cube root of a negative number is negative. Example: ∛(−8) = −2.
You can use prime factorisation, long division methods, or approximation techniques. The calculator provides instant results but learning the steps builds understanding.