Square Root, Cube Root and Nth Root Calculator
By Hesaplayıcı
The square root calculator finds the square root, cube root or any nth root of the number you enter. For a whole number it also simplifies the root to a√b: a is the coefficient taken out of the root and b is the number left under it. Enter the number and the index, then press Calculate.
Worked example: Square root of 72
Root
8.49
- Coefficient outside the root (a)
- 6
- Number under the root (b)
- 2
Step by step
Root
root = √x
root = √72 = 8.49
√x = a × √b, a² × b = x
√72 = √(6² × 2) = 6 × √2 = 8.49
How to use
- Enter the Number. A negative number works only with an odd index.
- Enter the Index of the root (n): 2 for a square root, 3 for a cube root, or another whole number up to 100.
- Press Calculate. You get the root and, for a whole number, the simplified radical.
Formula
ⁿ√x = a × ⁿ√b, where aⁿ × b = x. For a square root n = 2: √x = a × √b, a² × b = x.
x is the number you enter and n the index. a is the largest whole number whose nth power divides x. b is x divided by aⁿ, and it has no nth power factor left. For example, √50 = √(5² × 2) = 5 × √2 and ∛54 = ∛(3³ × 2) = 3 × ∛2. When the number is a perfect nth power, b = 1 and the root is a whole number.
Worked example
The worked example on this page takes the square root of 72 and simplifies it. The first step gives the square root; the second gives the coefficient taken out of the root and the number left under it.
Limits
The index is a whole number from 2 to 100; fractional indices are not included. Negative numbers have no real even root, so the calculator returns an error, and it gives no complex roots. Decimals are not simplified. The root is accurate to about 16 significant digits; the page may shorten it. The number can be from −10¹² to 10¹².
Frequently Asked Questions
What is a square root?
The square root of a number is the non-negative number whose square is that number. For example, 14² = 196, so √196 = 14.
How do you simplify a square root?
Write the number under the root as a perfect square times another number, then take the root of the perfect square out. For example, √50 = √(25 × 2) = 5√2.
How do you calculate a cube root?
The cube root is the number whose cube is the given number. Enter 3 as the index. For example, 4³ = 64, so ∛64 = 4; ∛54 simplifies to 3∛2.
Does a negative number have a root?
Not an even root (square root, fourth root) among the real numbers, because no real number has a negative square. An odd root exists and is negative: ∛−125 = −5.
Can I take the square root of a decimal?
Yes. The calculator gives the square root of a decimal, but it does not simplify it to a√b.
Calculation rules
- The n-th root is the number whose n-th power is the given number. A square root (n = 2) and every other even root is 0 or positive, and negative numbers have no real even root. Odd roots, such as the cube root, exist for negative numbers and are negative: ∛−125 = −5.
- For whole numbers the root is written as a × ⁿ√b: a is the largest whole number whose n-th power divides the number, and b has no n-th power factor left. Decimals are not simplified.
- The root is not rounded; it is accurate to about 16 significant digits (double precision). The root of a perfect power is given as a whole number. The page may shorten numbers.