Quick answer: √72 = 8.4852813742, which simplifies to 6√2 because 72 = 6² × 2; check: 8.4852813742² ≈ 72.
How it's calculated
The n-th root of x is the number that gives x when multiplied by itself n times: ⁿ√x = x^(1/n). The square root (n = 2) of 81 is 9 and the cube root (n = 3) of 125 is 5.
Odd roots of negative numbers are real: ∛−8 = −2 because (−2)³ = −8. Even roots of negative numbers are not real numbers.
To simplify a radical, split the number into prime factors and take out every factor that appears n times: √72 = √(2³ × 3²) = √(6² × 2) = 6√2. The prime factorization calculator shows that split, and the exponent calculator does fractional powers such as 8^(2/3).
Worked examples
- Square root of 72: √72 = 6√2 = 8.4852813742
- Cube root of 125: ∛125 = 5
- Cube root of a negative: ∛−8 = −2
Frequently asked questions
What is a perfect square?
A whole number whose square root is also whole: 1, 4, 9, 16, 25 and so on. Their roots need no simplifying.
Why does √−4 give an error?
No real number multiplied by itself gives a negative result. Its value 2i is a complex number, which this calculator does not show.
Is the square root of 9 equal to 3 or −3?
Both 3² and (−3)² are 9, but the √ symbol means the principal (non-negative) root, which is 3.
What is the square root of 72?
6√2, which is about 8.485. 72 is 36 × 2, and the 36 comes out as 6.
How can I estimate a square root in my head?
Find the perfect squares on either side. √50 sits between 7 (49) and 8 (64), right next to 7, so it's a shade over 7, about 7.07.
What is the cube root of 1,000?
10, because 10 × 10 × 10 = 1,000.