Exponent & Root Calculator

How to Use the Exponent & Root Calculator

Choose Exponent to calculate a base raised to a power, or Nth root to find a square root, cube root, or another whole-index root.

In Exponent mode, enter the base and exponent. Positive, zero, negative, and supported decimal exponents can be used when the result is real and within the calculator's numeric range.

In Root mode, enter the value and choose a square root, cube root, or custom positive whole-number index.

What Is an Exponent?

An exponent describes a power. In bⁿ, b is the base and n is the exponent.

For a positive whole-number exponent, the exponent tells you how many times the base appears as a factor. For example, 2⁴ = 2 × 2 × 2 × 2 = 16.

Exponents can also be zero, negative, or fractional, so exponentiation extends beyond repeated multiplication.

Positive Whole-Number Exponents

For a positive integer n, bⁿ means multiplying b by itself n times.

For example, 3⁴ = 3 × 3 × 3 × 3 = 81.

The sign of a negative base depends on the exponent. A negative base raised to an even whole-number exponent is positive, while a negative base raised to an odd whole-number exponent remains negative.

Zero Exponents

For any nonzero base b, b⁰ = 1.

For example, 5⁰ = 1 and (-12)⁰ = 1.

This calculator does not assign a value to 0⁰. That expression is treated as undefined here rather than automatically returning 1.

Negative Exponents

A negative exponent means take the reciprocal of the corresponding positive power.

For example, 2⁻³ = 1/2³ = 1/8 = 0.125.

A zero base cannot have a negative exponent because that would require division by zero.

Decimal and Fractional Exponents

Fractional exponents are closely connected to roots. For valid real-number cases, an exponent of 1/n corresponds to an nth root.

For example, 16⁰·⁵ = 4 because an exponent of 0.5 is equivalent to taking the square root of 16.

This calculator accepts decimal exponents for nonnegative bases when JavaScript can produce a finite real result. For negative bases, it deliberately requires a whole-number exponent rather than trying to interpret decimal input as an exact rational fraction.

Exponent Rules

Exponent rules can simplify expressions that use the same base.

When multiplying powers with the same base, add the exponents. When dividing them, subtract the exponents. Raising a power to another power multiplies the exponents.

These identities are useful algebraically, but the calculator itself evaluates the base and exponent you enter directly.

What Is an Nth Root?

The nth root of a number x is a value that produces x when raised to the nth power.

For example, the cube root of 125 is 5 because 5³ = 125. The fourth root of 81 is 3 because 3⁴ = 81.

The number n is called the root index. This calculator requires the index to be a positive whole number.

Square Roots

A square root uses index 2. The principal square root √x is the nonnegative number whose square equals x.

For example, √144 = 12 because 12² = 144.

Although both 12 and -12 solve the equation x² = 144, the radical symbol √144 refers specifically to the principal value 12.

Cube Roots

A cube root uses index 3. For example, ∛125 = 5 because 5³ = 125.

Unlike square roots, cube roots can be real for negative inputs. ∛(-27) = -3 because (-3)³ = -27.

This works because an odd power preserves the sign of its base.

Even and Odd Roots of Negative Numbers

A negative number has a real nth root when the index is odd. For example, the fifth root of -32 is -2.

A negative number does not have a real nth root when the index is even. There is no real number whose square, fourth power, or other even power is negative.

Because this calculator operates in the real-number system, it reports an error for even roots of negative values instead of returning a complex result.

Root Index 1

An index of 1 is valid. The first root of a number is simply the number itself.

For example, ¹√25 = 25 because 25¹ = 25.

It is rarely written explicitly, but it follows the same nth-root definition as other positive whole-number indices.

Roots and Fractional Exponents

For valid real-number cases, roots can be written using fractional exponents. The nth root of x corresponds to x^(1/n).

For example, ∛8 = 8^(1/3) = 2 and ⁴√16 = 16^(1/4) = 2.

The Root mode is preferable when you want to work explicitly with a whole-number root index, especially for negative values where odd and even indices behave differently.

Perfect Powers and Irrational Results

Some root calculations produce exact whole numbers. For example, √81 = 9 and ∛216 = 6.

Other values do not have an exact rational root. √2, for example, is irrational and must be represented numerically as an approximation.

The calculator displays a decimal result using the available floating-point precision.

Exponent Growth and Decay

Powers can grow or shrink very quickly. With a base greater than 1, increasingly large positive exponents usually produce rapidly growing results.

Negative exponents of a base greater than 1 move in the opposite direction because they produce reciprocals.

For example, 10³ = 1000 while 10⁻³ = 0.001.

Why Some Results Are Rejected

Some exponent and root expressions do not have a real result, while others exceed the numeric range available to the browser.

Examples rejected by this calculator include an even root of a negative number, zero raised to a negative exponent, and a negative base with a non-whole-number exponent.

Extremely large results can overflow to Infinity, while extremely tiny nonzero results can underflow to zero. The calculator reports these cases instead of presenting the overflowed or underflowed value as a valid answer.

Common Exponent and Root Mistakes

Treating a negative exponent as a negative result: b⁻ⁿ means 1/bⁿ; it does not simply change the sign.

Confusing -2² with (-2)²: Parentheses matter. (-2)² = 4, while the conventional interpretation of -2² is -(2²) = -4.

Assuming every negative number has a real square root: Even roots of negative numbers are not real.

Forgetting the root index: √x means a square root, while ∛x means a cube root.

Confusing a root with all solutions of an equation: √9 is 3, even though x² = 9 has solutions x = 3 and x = -3.

How This Calculator Works

Exponent mode evaluates the entered base raised to the entered exponent. It checks the real-number domain first, including the special cases involving zero and negative bases.

Root mode evaluates the nth root using the exponent 1/n. For negative values with an odd index, it calculates the root from the absolute value and restores the negative sign.

The calculator rejects nonfinite results and numeric underflow where a nonzero mathematical result would be represented as zero.

Numerical Precision

Results are calculated using JavaScript floating-point numbers. Many integers and simple powers are exact, while irrational roots and some decimal powers are necessarily approximate.

Very large exponents, large bases, or extremely small results can exceed the supported numeric range even when the mathematical expression itself is meaningful.

For symbolic work involving exact radicals, rational exponents, or complex-number branches, use the original expression rather than treating a rounded decimal result as exact.

Exponent and Root Formulas

Powers and roots are inverse operations in appropriate real-number domains.

$b$

Base

$m,n$

Exponents or whole-number indices, depending on the formula

$x$

Value whose root is being found

Examples

Frequently Asked Questions