What a root is
The square root of a number x is the number that, multiplied by itself, gives x: √49 = 7 because 7 × 7 = 49. More generally, the nth root of x is the number that gives x when raised to the power n. The cube root of 27 is 3, because 3 × 3 × 3 = 27. This calculator computes any root and tells you whether the result is exact.
The formulas
- nth root: ⁿ√x = x^(1/n)
- Square root: √x = x^(1/2)
- Cube root: ∛x = x^(1/3)
A root is the same as a power with a fractional exponent, which is how the calculator computes it. The check line raises the result back to the power n to confirm it returns the original number.
Worked example
Find the square root of 50.
- √50 ≈ 7.0710678.
- 50 lies between the perfect squares 49 and 64, so the root lies between 7 and 8.
- Since 50 = 25 × 2, the exact form is 5√2.
- The cube root of 50 is 3.6840315.
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 |
When the number is a perfect power, the calculator marks the result as exact.
Simplifying square roots
To simplify a square root by hand, look for the largest perfect square that divides the number: √72 = √(36 × 2) = 6√2, and √50 = √(25 × 2) = 5√2. The decimal value is the same; the simplified form is preferred in algebra because it stays exact.
Roots of negative numbers
An odd root of a negative number is a negative real number: ∛(−8) = −2. An even root of a negative number has no real value: there is no real number whose square is −4. In the complex numbers, √(−4) = 2i. The calculator flags this case.
Estimating a square root mentally
Find the nearest perfect square, then adjust: √50 ≈ 7 + (50 − 49) ÷ (2 × 7) = 7.071. This shortcut, based on Newton’s method, is accurate to three or four digits when the number is close to a perfect square.
Where roots are used
Square roots give the side of a square from its area, distances with the Pythagorean theorem, standard deviations in statistics and the root-mean-square value of an alternating current. Cube roots give the side of a cube from its volume, and higher roots appear in compound growth rates, such as the annual rate that doubles an investment in 10 years: ¹⁰√2 − 1 = 7.18%.