Root Calculator and Radical Simplifier
Enter a number and the root n, then press Calculate. n = 2 gives the square root, 3 the cube root, and any other n the nth root. A negative number has a real root when n is odd, so the cube root of −8 is −2. The square root of −4 gets a message, because it has no real value. The result shows up to 8 decimal places. For a whole number and a whole root, it also simplifies the radical, so √72 comes back as 6√2.
How to use the Root Calculator
- Enter your numbers in the Root Calculator above.
- Press the Calculate button to see the result.
- To try other numbers, change any value and press the button again.
The root definition
ⁿ√x = x^(1/n) · · · because (x^(1/n))^n = x
The nth root of x is the number that gives x when you raise it to the power n. The square root is the 2nd root and the cube root the 3rd. n does not have to be a whole number. The calculator works out x^(1/n) for any n other than 0, and a negative n gives 1 divided by the root, so n = −2 turns 4 into 0.5.
Worked example
√144 = 12 (because 12² = 144). ∛125 = 5 (because 5³ = 125). The 4th root of 81 = 3 (because 3⁴ = 81). The 5th root of 32 = 2 (because 2⁵ = 32). To simplify a radical, enter 72 with root 2. The calculator shows 8.48528137 and √72 = 6√2, because 72 = 6² × 2. Enter 54 with root 3 and it shows 3.77976315 and ∛54 = 3∛2.
Frequently asked questions
Can I take roots of negative numbers?
Yes, when n is an odd whole number. The cube root of −8 is −2, because (−2)³ = −8, and the calculator shows −2. An even root of a negative number, such as √−4, is not a real number. In complex numbers it is 2i. The calculator works with real numbers only, so it shows a message instead.
What is ⁿ√0?
0 for any positive n, because 0 raised to a positive power is 0. With a negative n the root of 0 would mean dividing by 0, so the calculator shows a message.
How do I simplify a radical?
Take out every perfect power. √72 = √(36 × 2) = 6√2, and ∛54 = ∛(27 × 2) = 3∛2. The calculator does this for any whole number up to 9,007,199,254,740,991 and any whole root of 2 or more, and it shows the decimal beside the radical. A prime such as 7 stays √7, because nothing comes out of it. −54 with root 3 gives −3∛2.
Why does a decimal get no simplified radical?
The form needs a whole number under the root, so 7.5 shows only its decimal root. The same goes for a root that is not whole, such as n = 2.5, and for a negative n. Write the number as a fraction by hand first if you need a radical from it: √7.5 = √(15/2).