Mathematics calculator

Radical Calculator

Calculate a real nth root using a whole-number degree from 1 through 100.

Calculator inputs

Enter your values

Real nth root
Calculated output

Result summary

Your result will appear here

Complete the required fields and select the calculate button.

An nth root and its inverse power relationship
Raising the calculated root to the entered degree should recover the radicand apart from display rounding.

What is Radical Calculator?

A radical asks for the number that produces a given value when raised to a chosen whole-number power. Square roots use degree 2, cube roots use degree 3, and the same idea extends to higher degrees. This calculator evaluates real nth roots and displays up to six decimal places.

Odd roots can accept negative numbers: the cube root of −125 is −5. Even roots of negative numbers are outside the real-number system, so the page rejects them instead of displaying a misleading value. The root degree must be a whole number from 1 through 100.

How to Use This Calculator

  1. Enter the number under the radical, also called the radicand.
  2. Enter the radical degree as a whole number. Use 2 for a square root or 3 for a cube root.
  3. Select Calculate. The page evaluates the principal real root and rounds the display to six decimal places.
  4. Check a simple result by raising the answer to the entered degree.

For a negative radicand, select an odd degree. An even degree has no real result, though complex-number methods can describe it. Degree 1 simply returns the original number.

Formula Used

y = ⁿ√x = x1/n, meaning yn = x

Here x is the radicand and n is a positive integer degree. For even n, the real principal root requires x ≥ 0. For odd n, a negative x has a negative real root. Decimal answers may be approximations when x is not a perfect nth power.

Variables Explained

Symbol or outputMeaning
xThe radicand, or number placed under the radical sign.
nThe positive whole-number degree, from 1 to 100 on this page.
yThe principal real nth root returned by the calculation.
Perfect nth powerA value whose nth root is an integer, such as 81 for degree 4.
Rounded resultAn approximate display to six decimal places when the root does not terminate neatly.

Worked Example

Take the cube root of −125. Enter −125 as the number and 3 as the degree. Since 3 is odd, a negative real answer is allowed. The magnitude is ∛125 = 5, and the sign remains negative, so the result is −5.

Check it by multiplying −5 three times: (−5) × (−5) × (−5) = −125. By comparison, entering −125 with degree 2 would be rejected because no real number squared equals −125.

Why Use This Calculator?

An integer answer usually signals a perfect power, such as the fourth root of 81 being 3. A decimal answer can still be valid; for example, √2 is approximately 1.414214 on this page. Keep more precision than the display provides if the root will be used in a sensitive later calculation.

The calculator returns the principal root. For √25, that principal value is 5, even though both 5 and −5 solve the equation y² = 25. That distinction matters when moving between evaluating a radical and solving an equation.

Roots undo powers, but only when the same degree is used. A square root reverses squaring on the nonnegative principal branch, while a cube root reverses cubing for positive and negative real values. Parentheses are useful when checking negative roots because −5³ and (−5)³ can be displayed differently by some tools even though the intended base here is −5.

Higher degrees can magnify the effect of display rounding during a check. If a six-place approximation is raised to the twentieth power, a small digit difference may become noticeable. That does not by itself mean the original root was calculated incorrectly.

Common Mistakes

  • Confusing the root degree with a multiplier. A cube root is not the same as dividing by three.
  • Entering zero or a decimal as the degree even though an nth-root index here must be a positive integer.
  • Trying to take an even real root of a negative number.
  • Assuming the principal square root includes both positive and negative signs.
  • Raising a rounded answer back to a high degree and expecting an exact match.

Frequently Asked Questions

What does the radical degree mean?

It tells how many equal factors reproduce the radicand. With degree 3, the returned number multiplied by itself three times gives the original number.

Can I calculate a square root?

Yes. Enter 2 as the degree. The result shown is the principal, nonnegative square root when the radicand is nonnegative.

Why is an even root of a negative number rejected?

No real number raised to an even power is negative. Such roots can be expressed with complex numbers, but this page is limited to real results.

Are odd roots of negative numbers allowed?

Yes. An odd number of negative factors has a negative product, so negative values have negative real odd roots.

Why do some answers have decimals?

Many numbers are not perfect nth powers. Their roots may be irrational or may simply require more digits than the page displays, so the result is rounded.

Is the first root useful?

Degree 1 returns the radicand itself because every number to the first power is unchanged. It is valid, though usually it does not simplify a problem.

How do I verify a radical result?

Raise the displayed answer to the chosen degree. Allow for a small difference when the displayed root was rounded to six decimal places.

Sources & References