Calculate the nth Root of a Number

What the Root Calculator Does and Who It's For

This Root Calculator finds the nth root of any number you enter. Set n to 2 for a square root, 3 for a cube root, or any other positive whole number to get a higher-order root such as the 4th or 5th root. It returns the principal real root, which is the standard single value most problems expect.

It's built for students checking algebra and geometry homework, engineers and tradespeople working with areas, volumes, and scaling, and anyone needing a quick, accurate answer without reaching for a spreadsheet. The nth root, cube root, and square root all use the same underlying operation, so one tool covers them all.

How the nth Root Works (The Formula)

An nth root reverses raising a number to the nth power. The nth root of a value is the number that, multiplied by itself n times, gives that value. In exponent form the formula is simply:

result = value^(1/n)

For example, the square root of x is x^(1/2) and the cube root of x is x^(1/3). The calculator computes value^(1/n) directly, which is why it handles any root index, not just 2 or 3.

• n = 2 -> square root: result = value^(1/2)
• n = 3 -> cube root: result = value^(1/3)
• n = 4 -> fourth root: result = value^(1/4)
• Check your answer by raising the result back to the power n; you should get the original value.

Worked Example

Suppose you want the cube root of 1728. Here n = 3 and value = 1728, so result = 1728^(1/3).

Because 12 x 12 x 12 = 1728, the cube root is exactly 12. You can confirm it by cubing the answer: 12^3 = 1728, which matches the input.

For a non-perfect example, the 4th root of 50 is 50^(1/4) = about 2.6591. Checking: 2.6591^4 = about 50.0, so the result is correct to the rounding shown.

Negative Numbers and Why Some Roots Have No Real Answer

The sign of the root depends on whether n is even or odd. With an odd index you can take the root of a negative number: the cube root of -27 is -3, because (-3) x (-3) x (-3) = -27.

With an even index, a negative input has no real root. There is no real number whose square equals -9, for instance, because any real number squared is zero or positive. In those cases the answer lives in the complex numbers rather than the real ones, and a real-root calculator will flag it as undefined.

Principal Root vs. Multiple Roots

Every positive number actually has two square roots, one positive and one negative; the square roots of 25 are +5 and -5. This calculator returns the principal root, which is the non-negative one (+5), matching the standard radical convention used in most coursework.

If your problem involves solving an equation like x^2 = 25, remember to include both the positive and negative solutions yourself. The calculator gives the principal value; the surrounding math may call for its opposite as well.

Tips and Common Mistakes

A few habits keep your results reliable and prevent the errors that trip people up most often.

• Don't confuse the root with division: the square root of 16 is 4, not 8 (16 / 2).
• Use a fractional power, not a negative one. value^(1/n) is the root; value^(-n) is something different entirely.
• Watch the index n. A larger n pulls results toward 1, so the 10th root of a number is much closer to 1 than its square root.
• Roots of perfect powers are exact; most other roots are irrational, so the displayed answer is rounded. Carry extra digits if you plan to use it in further calculations.
• Always verify by raising your result to the power n and comparing it with the original value.