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Exponent Calculator

Apply exponent rules — product, quotient, power, zero, negative, fractional and the powers of products and quotients — and convert between decimal, scientific and engineering notation. Live updates as you type.

Exponent expression

Updates as you type
Mode
What do you want to do? ?
Rule
Which rule should we apply? ?
Quick presets (click to fill)
Inputs
Base (a) ? —
−10−50510
Exponent m —
−10−50510
Exponent n —
−10−50510

Growth of aᵏ

Visualizes how the base behaves as the exponent grows
y = aᵏ

Values table

k aᵏ Scientific Magnitude

Formula

am × an = am+n
Product: am × an = am+n
Quotient: am ÷ an = am−n
Power of power: (am)n = am·n
Zero: a0 = 1  (a ≠ 0)
Negative: a−m = 1 / am
Fractional: am/n = n√(am)
Power of product: (a·b)m = am · bm
Power of quotient: (a/b)m = am / bm
Scientific notation: a × 10n  (1 ≤ |a| < 10)
a, b
Bases — the numbers being raised to a power.
m, n
Exponents — the powers applied. May be positive, negative or fractional.
Worked example — your numbers
  1. Rule: —
  2. Setup: —
  3. Combine exponents: —
  4. Result: —

Negative bases combined with non-integer exponents can produce complex numbers — only real results are shown. By convention 00 = 1 in combinatorics, though it is left undefined in analysis.

Examples

How It Works

Exponents are a compact way to write repeated multiplication: aⁿ means the base a multiplied by itself n times. The eight rules in this calculator describe how exponents combine when you multiply, divide, raise to another power, or distribute across products and quotients.

The Product rule (aᵐ × aⁿ = aᵐ⁺ⁿ) and Quotient rule (aᵐ ÷ aⁿ = aᵐ⁻ⁿ) work whenever both terms share the same base — you simply add or subtract the exponents. The Power of a power rule, (aᵐ)ⁿ = aᵐ·ⁿ, multiplies the exponents. Zero exponents always evaluate to 1 (with a ≠ 0 by convention), and negative exponents represent reciprocals: a⁻ⁿ = 1/aⁿ.

Fractional exponents bridge powers and roots: aᵐ⁄ⁿ is the n-th root of aᵐ. Fractional exponents on negative bases can produce complex numbers — this calculator only shows real results. The power of a product and power of a quotient rules distribute an outer exponent across multiplication and division: (a·b)ᵐ = aᵐ · bᵐ.

Scientific notation writes any number as a × 10ⁿ where 1 ≤ |a| < 10 — useful for very large or very small magnitudes. Engineering notation rounds n to a multiple of three so the coefficient lines up with SI prefixes (kilo, mega, giga, milli, micro, nano).

Tips & Best Practices

The Product, Quotient and Power of a power rules only work when the bases match. If they don't, you have to evaluate each side independently.
A negative exponent never makes the value negative — it makes it a reciprocal. (−2)³ = −8, but 2⁻³ = 1/8.
Fractional exponents are the same as roots: a^(1/2) is √a, a^(1/3) is ∛a, and a^(2/3) is the cube root of a².
Use scientific notation for any number with more than ~6 digits or fewer than 4 leading decimal zeros — it's easier to read and to compare magnitudes at a glance.
Engineering notation maps directly to SI prefixes (10³ = kilo, 10⁶ = mega, 10⁻³ = milli, 10⁻⁶ = micro). Useful for electronics, physics and astronomy.

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Frequently Asked Questions

What is an exponent?

An exponent tells you how many times to multiply a number (the base) by itself. In 2⁵ the base is 2 and the exponent is 5, so 2⁵ = 2 × 2 × 2 × 2 × 2 = 32.

Look at the Quotient rule: aⁿ ÷ aⁿ = aⁿ⁻ⁿ = a⁰. But aⁿ ÷ aⁿ also equals 1, so a⁰ must equal 1 for any non-zero base. The expression 0⁰ is conventionally taken as 1 in combinatorics and polynomial expansion, though it is left undefined in analysis.

Yes. (−2)³ = −8 and (−2)⁴ = 16 — odd exponents preserve the sign and even exponents make it positive. Things get tricky with fractional exponents, because (−1)^(1/2) is the imaginary unit i. This calculator only displays real results.

Scientific notation writes a number as a × 10ⁿ where the coefficient a satisfies 1 ≤ |a| < 10. It compresses very large or very small numbers — Avogadro's constant is 6.022 × 10²³ rather than a 24-digit decimal.

Engineering notation forces the exponent to be a multiple of 3, so the coefficient sits between 1 and 1000. This lines up with SI prefixes: 4.5 × 10³ Hz is 4.5 kHz, 22 × 10⁻⁶ F is 22 μF. Scientists, electrical engineers and physicists prefer it for that reason.

e ≈ 2.71828 is Euler's number, the base of the natural logarithm. eˣ is the unique exponential function whose derivative equals itself, which is why it appears throughout calculus, statistics and continuous-compounding finance.