xⁿ Exponent Calculator
Calculate powers, roots, and logarithms with full exponent rules.
How to Use This Calculator
This calculator handles four types of calculations: raising a base to a power, taking any root (square root, cube root, nth root), computing logarithms in any base, and applying the rules of exponents step by step. Choose the mode you need and fill in the values.
For a power (x^n), enter the base and the exponent. You can use decimals and negatives for both.
For a root, enter the number and the root degree. Degree 2 is a square root, degree 3 is a cube root.
For a logarithm, enter the value and the base. Base 10 is the common log; base e (2.718...) is the natural log.
Check the step-by-step section to see which exponent rule was applied and how the answer was reached.
Laws of Exponents
These six rules cover every exponent situation you will come across. The product rule is the most used: when you multiply same-base numbers, you just add the exponents. The negative exponent rule trips people up most often: 2^(-3) does not mean negative 8, it means 1 divided by 2^3, which equals 1/8 or 0.125.
Worked Examples
Where Exponents Come Up
Exponents are everywhere in finance, science, and computing. Compound interest uses the formula A = P x (1 + r)^n, where n is the number of periods. A 7% annual return on $1,000 over 20 years gives $1,000 x (1.07)^20 = $3,870. Exponential growth and decay equations describe population growth, radioactive decay, and the cooling of hot objects.
In computing, binary numbers are powers of 2: 1 KB is 2^10 = 1,024 bytes, 1 MB is 2^20 = 1,048,576 bytes. When you work with logarithms in programming (sorting algorithms, information theory, audio levels in decibels), you are always working with exponent relationships. The natural logarithm (base e) shows up in probability, statistics, and any continuous growth model.
Frequently Asked Questions
What is an exponent?
An exponent (or power) tells you how many times to multiply a base by itself. 2³ = 2×2×2 = 8. The base is 2 and the exponent is 3.
What is a negative exponent?
A negative exponent means the reciprocal: a^(-n) = 1/a^n. Example: 2^(-3) = 1/8 = 0.125.
What is a fractional exponent?
A fractional exponent represents a root: a^(1/n) = nth root of a. a^(m/n) = nth root of a^m = (nth root of a)^m. Example: 8^(2/3) = cube root of 8² = 4.
What is the difference between log and ln?
log₁₀ (common log) uses base 10. ln (natural log) uses base e (≈2.71828). ln is fundamental in calculus and exponential growth models.
What are the laws of exponents?
Product: aᵐ × aⁿ = a^(m+n). Quotient: aᵐ/aⁿ = a^(m-n). Power: (aᵐ)ⁿ = a^(mn). Zero: a⁰=1. Negative: a^(-n)=1/aⁿ.