Can I raise a negative base to a fractional exponent?

Only if the fraction’s denominator is odd (e.g., (−8)^(1/3) = −2). If the denominator is even, there is no real result.

How do you handle decimal exponents?

For a > 0, a^x = e^(x ln a). If a ≤ 0 and x is not a rational number with an odd denominator, the real-valued result is undefined.

What exact forms do you show?

For rational exponents p/q we display a^(p/q) as √[q]{a^p}. When the result is an integer (e.g., 27^(2/3) = 9), we report the exact integer.

What precision is used for decimals?

You can choose from 0 to 15 decimal places. Scientific notation uses the same number of digits.

Do you support scientific notation inputs?

Yes. You can enter numbers like 3e2 or 3×10^2. Fractions such as 1/3 are supported for the exponent as well.

Is this calculator accurate for very large exponents?

Results follow IEEE-754 double precision. Extremely large or small values may overflow to Infinity or underflow to 0.

Exponent Calculator | Math Algebra Tool with Steps

Data Source and Methodology

Authoritative Data Source: NIST Digital Library of Mathematical Functions (DLMF), Chapter 1: Elementary Functions, Section 1.2 and 1.8. Latest edition and updates available online. https://dlmf.nist.gov/1.2

All computations strictly follow the formulas and conventions provided by this source.

The Formula Explained

For integer n:

$$ a^n = \underbrace{a \cdot a \cdot \dots \cdot a}_{n\ \text{times}} \quad \text{and} \quad a^{-n} = \frac{1}{a^n},\ a \neq 0. $$

For rational exponent x = p/q in lowest terms (q > 0):

$$ a^{p/q} = \big(\sqrt[q]{a}\big)^{p} = \sqrt[q]{a^p}. $$

For real exponent x and a > 0:

$$ a^{x} = e^{x \ln(a)}. $$

Glossary of Variables

a (Base): The number to be raised to a power.
x (Exponent): The power applied to the base; can be integer, negative, decimal, or a fraction p/q.
Exact form: Symbolic expression for rational exponents (e.g., √[3]{a^2}) when applicable.
Decimal value: Numerical approximation displayed with your selected precision.
Scientific notation: Exponential notation using a×10^b (shown as aEb for machine readability).
Precision: Number of digits after the decimal point (0–15).

How It Works: A Step-by-Step Example

Problem: Compute 27^(2/3).

Identify p/q: x = 2/3 with p = 2 and q = 3 (q is odd, domain OK for a = 27).
Use the rational rule: a^(p/q) = (√[q]{a})^p. So 27^(2/3) = (√[3]{27})^2.
Compute the root: √[3]{27} = 3 because 3^3 = 27.
Raise to power p: 3^2 = 9.
Answer: 27^(2/3) = 9 exactly. Decimal = 9.0000000000; Scientific = 9.0000000000e+0.

Frequently Asked Questions (FAQ)

What inputs are supported?

Base accepts integers, decimals, and scientific notation (e.g., 3e2 or 3×10^2). Exponent accepts integers, negatives, decimals, and fractions like 1/3.

How do you handle negative bases?

If the exponent is a rational p/q, a negative base is valid only when q is odd. For decimal exponents that are not rational with an odd denominator, the real result is undefined.

Why do I see Infinity or 0?

JavaScript uses IEEE-754 double precision. Extremely large values overflow to Infinity; extremely small values may underflow to 0.

Can I copy the result?

Yes. Use the “Copy value” button under the results to copy the current numeric output.

What is the maximum precision?

You can select up to 15 decimal places. More digits would not be meaningful in double precision.

Do you show symbolic simplifications?

We display exact forms for rational exponents. When the result is an integer (e.g., 16^(1/2) = 4), the exact integer is shown.

Is complex arithmetic supported?

No. This tool focuses on real-valued results. If a real result does not exist for the given inputs, the calculator will indicate a domain issue.

Tool developed by Ugo Candido. Content verified by CalcDomain Editorial Team.
Last reviewed for accuracy on: September 14, 2025.