Exponent & Root Calculator (Powers, Square & Nth Roots)

Power

The root block below works independently of the power block.

Base

ExponentNegative and fractional values are allowed

Root: the number under the radical

Root: index (n)2 = square root, 3 = cube root

RESULT

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Scientific notation—

Reciprocal (base⁻ᵉ)—

Square of the base—

Cube of the base—

Square root of the base—

Digits in the result—

Root block

nth root—

Square root—

Cube root—

As a fractional exponent—

Check — root raised back—

Exact?—

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Negative exponents flip, they do not negate

2⁻³ is 1/8, not −8. A negative exponent means "the reciprocal of the positive power". The rule falls straight out of dividing powers: 2³ / 2⁶ = 2⁻³, and the left side is obviously 8/64 = 1/8. This is the single most common exponent mistake and it survives well into university-level work.

Fractional exponents are roots

x^1/n is the nth root of x. x^m/n is the nth root raised to the mth power — and you can do those two steps in either order. 8^2/3 is either the cube root of 64 or the square of 2; both give 4. Taking the root first usually keeps the numbers smaller and the rounding cleaner.

Why anything to the power of zero is 1

Follow the pattern down: 2⁴ = 16, 2³ = 8, 2² = 4, 2¹ = 2 — each step divides by 2. One more step gives 2⁰ = 1. The same argument in reverse is why 0⁰ is left undefined in most contexts: the pattern gives 1 but the sequence 0¹, 0² gives 0, and the two answers cannot both be right.

Negative bases and odd roots

The cube root of −27 is −3, because a negative times a negative times a negative is negative. But the square root of −27 is not a real number at all — no real number squared is negative. Odd roots of negatives work, even roots do not, and this calculator says so rather than silently returning NaN.

Frequently Asked Questions

What does a negative exponent mean?

It means take the reciprocal: x⁻ⁿ = 1/xⁿ. So 5⁻² is 1/25 = 0.04. It never produces a negative result unless the base itself is negative.

How do I calculate an nth root?

Raise the number to the power 1/n. The fifth root of 32 is 32^0.2 = 2. Enter the radicand and the index in the root block above and both the value and the check are shown.

Can I take the square root of a negative number?

Not within the real numbers. Even roots of negatives require imaginary numbers, so the calculator reports this rather than returning a meaningless value. Odd roots of negatives — cube root, fifth root — are perfectly real.

Why does my result show as 1.024e+3 instead of 1024?

That is scientific notation, used automatically when a number is too long or too small to display cleanly. The plain value is shown in the main result field; the scientific form is there for very large or very small answers.

Where these numbers come from

Sources

Official publications only. Links open the original document in a new tab.

National Institute of Standards and Technology Special Publication 811 — Guide for the use of the SI Significant figures and rounding conventions
National Institute of Standards and Technology CODATA fundamental physical constants Values of the fundamental physical constants

Exponent & Root Calculator