Exponent Calculator
Calculate a base number raised to any power, including negative and fractional exponents.
Comment ça marche
This calculator computes base^exponent directly — a base number multiplied by itself as many times as the exponent specifies, for whole-number exponents, and extended consistently to negative and fractional exponents as well.
A whole-number exponent simply means repeated multiplication: 2^4 = 2 × 2 × 2 × 2 = 16. Any nonzero number raised to the power of 0 equals 1 by definition (a mathematical convention that keeps exponent rules consistent), and any number raised to the power of 1 equals itself.
Negative exponents represent the reciprocal of the positive-exponent result: base^(−n) = 1 / (base^n). For example, 2^(−3) = 1/(2^3) = 1/8 = 0.125 — the negative sign flips the result to a fraction rather than making it negative.
Fractional exponents are equivalent to taking a root: base^0.5 is the square root of base, base^(1/3) is the cube root, and more generally base^(1/n) is the nth root. A fractional exponent like 1.5 combines both a power and a root: base^1.5 equals base^1 multiplied by base^0.5, so 4^1.5 = 4 × √4 = 4 × 2 = 8.
Exponents grow extremely fast for bases greater than 1 — this is the mathematical basis of "exponential growth," seen in compound interest, population growth, and viral spread, where small increases in the exponent produce dramatically larger results (2^10 = 1,024, but 2^20 is already over a million).
A negative base raised to an exponent follows its own sign rule: a negative base raised to an even exponent produces a positive result (since the negative signs cancel in pairs), while a negative base raised to an odd exponent stays negative — (−2)^2 = 4, but (−2)^3 = −8.
Exponent rules combine predictably when working with multiple powers of the same base: multiplying same-base powers adds their exponents (2² × 2³ = 2⁵ = 32), and raising a power to another power multiplies the exponents ((2²)³ = 2⁶ = 64) — these shortcuts let complex expressions be simplified without expanding every multiplication individually.
What does a fractional exponent mean?
A fractional exponent like 0.5 is equivalent to taking a root — base^0.5 is the square root of base.
What does a negative exponent mean?
It represents the reciprocal of the positive-exponent result: base^(−n) equals 1 divided by base^n, turning the result into a fraction.
What is any number raised to the power of 0?
1, by mathematical convention — this holds for any nonzero base and keeps exponent rules consistent across calculations.
Why does a negative base squared give a positive result?
Multiplying two negative numbers together produces a positive result, so an even exponent (which pairs up the negative signs) always yields a positive answer for a negative base.
What happens with a negative base and an odd exponent?
The result stays negative, since an odd number of negative signs multiplied together doesn't fully cancel out — for example, (−2)³ = −8.
Why do exponents grow so quickly?
Each additional increment to the exponent multiplies the previous result by the base again, causing the total to compound rapidly — this is the mathematical basis of exponential growth patterns.
How is a fractional exponent like 1.5 interpreted?
It combines a whole-number power with a root — base^1.5 equals base¹ multiplied by base^0.5 (the square root of base).
Can the exponent itself be a decimal or negative number in this calculator?
Yes — this calculator supports any real-number exponent, including negative, zero, and fractional values, not just positive whole numbers.
What happens when I multiply two powers with the same base?
Their exponents add together: 2² × 2³ = 2^(2+3) = 2⁵ = 32, a shortcut that avoids needing to expand and multiply each power separately.
What happens when I raise a power to another power?
The exponents multiply together: (2²)³ = 2^(2×3) = 2⁶ = 64, another shortcut rule for simplifying nested exponent expressions.
What happens when I divide two powers with the same base?
Their exponents subtract: 2⁵ ÷ 2² = 2^(5−2) = 2³ = 8 — the same underlying pattern as the multiplication rule, just in reverse.
Why is understanding exponent rules useful beyond a single calculation?
They let you simplify complex expressions with multiple powers of the same base without needing to expand every multiplication individually, which is especially useful in algebra and scientific notation.