Compute powers, negative and fractional exponents, nth roots and e^x — with full step-by-step working and rule reference.
Results update live as you type.
Result
Scientific
1.024e+3
e-notation
Sign
Positive
result polarity
Magnitude
3
order of 10
Inverse
9.77e-4
b−n = 1/bn
| Rule | Formula | Example |
|---|---|---|
| Product | bm · bn = bm+n | 2² · 2³ = 2⁵ = 32 |
| Quotient | bm / bn = bm−n | 2⁵ / 2² = 2³ = 8 |
| Power | (bm)n = bm·n | (2²)³ = 2⁶ = 64 |
| Zero | b0 = 1 | 5⁰ = 1 |
| Negative | b−n = 1/bn | 2⁻³ = 1/8 = 0.125 |
| Fractional | b1/n = ⁿ√b | 91/2 = √9 = 3 |
The Method
An exponent is repeated multiplication: b^n means multiply
b by itself n times. For integer exponents this is a direct
product. For negative exponents the result is the reciprocal of the
positive power: b−n = 1 / bn. For
fractional exponents, b1/n equals the nth root of
b. The natural exponential ex uses Euler's constant
e ≈ 2.71828 and is the basis for continuous growth, calculus, and
probability.
Working for 210
About This Tool
An exponent calculator evaluates expressions of the form bn — a base raised to a power. Enter any base and any exponent (including negative or fractional values) and the calculator returns the exact decimal result, scientific notation, magnitude, and the multiplicative inverse b−n.
Exponents condense repeated multiplication and are foundational across mathematics. They appear in scientific notation (encoding very large or very small numbers), compound interest formulas, radioactive decay, computer science complexity (logarithms and powers of two), polynomial algebra, and the natural exponential ex at the heart of calculus.
This calculator supports three modes: the standard bn power, the nth root (inverse of an exponent — ⁿ√a is the value x such that xn = a), and the natural exponential ex. All computations use IEEE 754 double-precision floating-point — accurate to about 15-17 significant digits.
Use this free exponent calculator for homework, exam revision, scientific notation conversions, compound-growth estimates, or as a quick reference for the rules of exponents. Every calculation runs locally in your browser — no sign-up, no tracking.
bn Power
Compute any positive, negative, or fractional power of any base.
nth Root
Inverse of an exponent — square root, cube root, or any nth root.
ex Mode
Natural exponential with Euler's number e ≈ 2.71828.
Growth Chart
Visualise how powers of b grow for k = 1..10 on the same scale.
100% Free & Private
No account, no tracking — every calculation runs locally in your browser.
Step-by-Step Working
See the expanded multiplication and scientific-notation result side by side.
Three modes, one workflow — choose the calculation you need and the answer appears instantly.
Use the tabs to switch between bn power, nth root, and ex. Each mode shows only the inputs relevant to that calculation.
Type any real base and any exponent — including negative and fractional values. The sliders give quick scanning through nearby values.
The headline shows the result in standard notation. The stats tile shows the scientific notation, sign, magnitude, and the inverse b−n.
The bar chart shows bk for k = 1..10 on the same relative scale, so you can see how dramatically exponential growth accelerates.
The exponent rules table shows the product, quotient, power, zero, negative, and fractional rules with worked examples — handy for homework and exam revision.
The formula card shows the expansion (e.g. 210 = 2 × 2 × ...) and the final value — the exact steps you would write out by hand.
Everything you need to know about exponents, powers, roots, and the natural exponential.
An exponent (or power) tells you how many times to multiply a number — the base — by itself. In the expression 2³, the base 2 is multiplied by itself 3 times: 2 × 2 × 2 = 8. We say "2 to the power of 3" or "2 cubed". An exponent of 2 is called a square, an exponent of 3 a cube.
A negative exponent represents the reciprocal of the positive power: b−n = 1 / bn. For example, 2−3 = 1 / 2³ = 1/8 = 0.125. Negative exponents appear naturally in scientific notation (e.g. 1.6e−19 for the charge of an electron) and in unit conversions (m·s−1 means metres per second).
A fractional exponent represents a root: b1/n = ⁿ√b. So 91/2 = √9 = 3, and 271/3 = ∛27 = 3. More generally, bm/n = (ⁿ√b)m. Fractional exponents are how mathematicians extend the idea of "power" beyond whole numbers and make functions like x0.5 meaningful.
Any non-zero base raised to the power of 0 equals 1: b0 = 1. This follows from the quotient rule: bn / bn = bn−n = b0, and any non-zero number divided by itself is 1. The value 00 is considered indeterminate in calculus but is often defined as 1 in algebra and combinatorics.
ex is the natural exponential function, where e ≈ 2.71828 is Euler's number. It is the unique exponential whose derivative equals itself: d/dx (ex) = ex. Because of this property, ex appears throughout calculus, continuous compounding, radioactive decay, probability (the normal distribution), and many physics and engineering models.
Use the product rule: bm · bn = bm+n. Add the exponents and keep the same base. For example, 2² · 2³ = 2⁵ = 32. The intuition is that multiplying b by itself m times, then n more times, is the same as multiplying it m + n times in total.
Use the power rule: (bm)n = bm·n. Multiply the exponents. For example, (2³)² = 2⁶ = 64. This is different from bmn (which is b raised to "m to the n") — order matters when exponents are stacked.
The nth root of a, written ⁿ√a, is the value x such that xn = a. The 2nd root is the square root (√), the 3rd is the cube root (∛). For positive a, ⁿ√a is uniquely defined. For negative a, odd roots exist as real numbers (∛−8 = −2) but even roots are complex.
Because each step multiplies by the base rather than adding. 210 = 1024, 220 ≈ 1 million, 230 ≈ 1 billion — doubling each step ten times multiplies by about 1000. This explosive growth is why exponential models describe compound interest, viral spread, and Moore's law.
The calculator uses IEEE 754 double-precision floating-point, with a range of roughly ±1.8 × 10308 and 15-17 significant digits. Beyond that, results overflow to ±Infinity. For exact arithmetic on huge integers, use the Big Number Calculator.
Scientific notation writes a number as a × 10n where 1 ≤ |a| < 10 and n is an integer. So 1024 = 1.024 × 10³ and 0.00056 = 5.6 × 10−4. It is the standard way to express very large or very small numbers compactly — atomic masses, astronomical distances, and physical constants are nearly always given this way.
No. Exponentiation is not commutative: 2³ = 8 but 3² = 9. The base and the exponent play different roles. The only solutions to ab = ba with a ≠ b are a = 2, b = 4 (or vice versa): 24 = 42 = 16. Otherwise, swapping base and exponent gives a different result.