√ Square Root Calculator
Find square roots, cube roots and any nth root to 10 decimal places, with the simplified radical form for whole numbers.
Quick answer: √72 = 6√2 ≈ 8.4852813742, because 72 = 36 × 2 and √36 = 6. Check: 8.4852813742² ≈ 72.
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Square Root Calculator inputs
Result
Square root
√72 ≈ 8.4852813742
Simplified: √72 = 6√2
- Result
- 8.4852813742
- Simplified radical
- 6√2
- Perfect power?
- No
Step by step
- Prime factorization: 72 = 2³ × 3²
- Take out every group of 2 equal factors: √72 = 6√2
- √72 ≈ 8.4852813742
- Check: 8.4852813742² ≈ 72
What is a square root?
The square root of x is the number that, multiplied by itself, gives x. More generally, the nth root of x is the number that gives x when raised to the power n.
√x = x^(1/2) · ⁿ√x = x^(1/n) · √(a × b) = √a × √b
Simplifying radicals
To write a square root in simplest radical form, split the number into a perfect square times what is left over:
- Find the prime factorization: 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3².
- Pair up equal factors: (2 × 2) and (3 × 3), with one 2 left over.
- Each pair comes out of the root as a single factor: 2 × 3 = 6.
- Result: √72 = 6√2 ≈ 8.4852813742.
For cube roots you group equal factors in threes instead of pairs: ∛54 = ∛(27 × 2) = 3∛2.
Perfect squares and cubes
| n | n² | n³ |
|---|---|---|
| 2 | 4 | 8 |
| 3 | 9 | 27 |
| 4 | 16 | 64 |
| 5 | 25 | 125 |
| 10 | 100 | 1,000 |
| 12 | 144 | 1,728 |
Estimating without a calculator
Find the perfect squares on either side. √50 lies between √49 = 7 and √64 = 8, much closer to 7, so it is a little over 7 (actually 7.0710678…). Memorising squares up to 15² makes this fast.
Tips
- Even roots (square, 4th…) of negative numbers are not real; odd roots of negatives are negative.
- Check any root by raising the answer back to the power n – the calculator shows this check.
- Simplified radicals are exact, so prefer them in algebra and geometry answers.
Frequently asked questions
How do you simplify a square root?
Factor the number and pull out any perfect-square factor. 72 = 36 × 2, so √72 = √36 × √2 = 6√2. Using the prime factorization 2³ × 3², each pair of equal primes comes out as one factor.
Can you take the square root of a negative number?
Not within the real numbers – no real number squared gives a negative. In complex numbers √−4 = 2i, where i = √−1. Odd roots of negatives are fine: the cube root of −27 is −3.
What is a perfect square?
A whole number whose square root is also a whole number: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 and so on. Their roots are exact; every other positive integer has an irrational square root.
How do I find a cube root or a 4th root?
Change the Root (index) box to 3 for a cube root, 4 for a fourth root, and so on. The nth root of x is the number that gives x when multiplied by itself n times – ∛125 = 5 because 5 × 5 × 5 = 125.