Root Calculator
The n-th root of a number is the value that, multiplied by itself n times, gives the number. The square root of 49 is 7 and the cube root of 27 is 3. This calculator finds any root, checks its own answer by raising it back to the power, and writes whole-number square and cube roots in simplified radical form.
How to use the root calculator
- Enter the number under the root sign.
- Enter the degree: 2 for a square root, 3 for a cube root, and so on.
- Read the value, the check and (for whole numbers) the simplified radical such as 6√2.
Formula
ⁿ√x = x1/n. To simplify a square root, pull out the largest perfect-square factor: √72 = √(36 × 2) = 6√2. For a cube root, pull out perfect cubes: ∛54 = ∛(27 × 2) = 3∛2.
Worked examples
A simplified radical
√72 ≈ 8.485. Since 72 = 36 × 2, it simplifies to 6√2.
A cube root
∛27 = 3 because 3 × 3 × 3 = 27. And ∛(−8) = −2, because odd roots of negatives are real.
Roots and radicals
Perfect squares and cubes to know
| Squares | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 |
|---|---|---|---|---|---|---|---|---|---|---|
| Root | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
Cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000.
Simplifying a square root
Find the largest perfect square that divides the number. √72: 36 divides 72, so √72 = √36 × √2 = 6√2. √50 = √25 × √2 = 5√2. Simplified radicals are the standard form in algebra because they are exact; the decimal 8.485… is only an approximation.
Estimating a root
To estimate √50, note that 7² = 49, so √50 is just over 7. A closer guess is 7.07. This kind of estimate helps you check whether a calculator answer is sensible.
Roots and exponents
Roots are exponents in disguise: √x = x1/2, ∛x = x1/3. That is why root rules mirror exponent rules. See exponents and powers explained.
Practice problems
Try these yourself first, then check your answer. The answers come from the calculator above.
- Simplify √50.
Show answer
2-th root of 50: 7.071067812
- What is the cube root of 125?
Show answer
3-th root of 125: 5
Common mistakes to avoid
- Expecting a real result from an even root of a negative number. √(−4) is not real; it is 2i.
- Assuming √(a + b) = √a + √b. It does not: √(9 + 16) = 5, not 3 + 4.
- Forgetting that every positive number has two square roots (±), though the radical sign means the positive one.
Frequently asked questions
What is a square root?
A number that gives the original when multiplied by itself. The square root of 49 is 7.
Can you take the root of a negative number?
Odd roots, such as cube roots, of negative numbers are real. Even roots are not real.
How do I simplify a radical?
Factor out the largest perfect power. √50 = √(25 × 2) = 5√2.
What is the fourth root of 16?
2, because 2 × 2 × 2 × 2 = 16.
Reviewed September 26, 2026 by the SolveCalcPro editorial team. Found a mistake? Tell us; see our editorial policy.