Radical Calculator
Compute the nth root of a number and, for square roots, simplify it to exact radical form a√b by pulling out perfect squares.
Calculator
- Any root index ≥ 2
- Square-root simplification a√b
- Odd roots of negatives
- Power-back check
Result & calculation receipt
- Method
- Square root with perfect-square extraction
- Formula
ⁿ√r = r^(1/n); √(k²·m) = k√m- Substitution
- √72 = 6√2 = 8.4852813742
- Steps
- Factor out perfect squares: √72 = √(36·2) = 6√2.
- Verification
- 8.48528137^2 = 72 ≈ 72.
- Exact result
- 6√2
- Decimal result
- 8.4852813742
- Engine
- math-solvers v1.4.0
How this calculator works
For a square root the calculator factors out the largest perfect square, so a number like 72 becomes 6√2. For other roots it evaluates the radicand to the power 1/n. It allows odd roots of negative numbers (which are real), rejects even roots of negatives (which are not real), and verifies the result by raising it back to the nth power.
Worked example
√72 = √(36 × 2) = 6√2, whose decimal value is about 8.485. The cube root of 27 is 3, and the cube root of −8 is −2, because (−2)³ = −8. An even root of a negative number, like √−4, is refused with a domain error and a pointer to complex numbers.
Key insight
Simplifying a radical to a√b keeps it exact — 6√2 is precise where 8.485 is only rounded. Radicals and fractional exponents are the same thing: the nth root of x is x^(1/n), which is why the two calculators agree.
Frequently asked questions
How do I simplify a square root?
Factor the radicand into a perfect square times a remainder and take the square root of the perfect square out front. So √72 = √(36 × 2) = 6√2. The calculator shows the perfect square it extracted.
Can I take the cube root of a negative number?
Yes. Odd roots of negatives are real: the cube root of −8 is −2 because (−2)³ = −8. The calculator handles any odd index.
Why is the square root of a negative number rejected?
An even root of a negative number is not a real number — √−4 is imaginary. The calculator returns a domain error and points to the complex-number tools.
How is an nth root related to an exponent?
The nth root of x equals x^(1/n). So a radical is just a fractional power, which is why this and the exponent calculator give matching answers.
How is the result checked?
By raising it back to the nth power and confirming it returns the radicand, which is shown in the receipt.
Methodology & verification
For index 2 the square root is simplified to exact form a√b by extracting the largest perfect-square factor; for other indices the value is radicand^(1/n). The index must be an integer ≥ 2. Even roots of negative radicands return a domain error (not real); odd roots of negatives are computed as negative reals. The result is verified by raising it to the nth power and confirming it recovers the radicand, and the exact and decimal forms are reported separately.
Engine: math-solvers v1.4.0 · Last reviewed: · Every result is computed client-side and never sent to a server. See the verification methodology and the scientific calculator validation.