Exponent calculator,
any base, any power.

Field guide

Everything you need to know about exponents and powers.

An exponent (or power) tells you how many times to multiply a number, called the base — by itself. Written asbⁿ, the base is b and the exponent is n. The expression 2⁵ means 2 × 2 × 2 × 2 × 2 = 32. Simple enough for integers, but exponents extend far beyond whole numbers — into negatives, fractions, decimals, and irrational numbers and understanding each case unlocks a huge swath of algebra, calculus, and applied science.

Positive integer exponents

The most intuitive case: repeat the multiplication n times.

Examples: 3^4 = 81, 10^6 = 1 000 000, 2^10 = 1 024. Larger exponents grow surprisingly fast —2^64 ≈ 1.84 × 10^19, exceeding the number of seconds in the age of the universe.

The zero exponent: b⁰ = 1

Any non-zero base raised to the power 0 equals 1. This can be proven from the quotient law: b^n / b^n = b^(n−n) = b^0, and any number divided by itself equals 1 (when the denominator is non-zero). The case 0^0 is technically indeterminate in calculus, but by convention equals 1 in combinatorics and most programming contexts.

Negative exponents: reciprocals

A negative exponent means take the reciprocal of the positive power:

Examples: 2^(−3) = 1/8 = 0.125, 10^(−4) = 0.0001, 5^(−1) = 0.2. Negative exponents are the foundation of scientific notation for very small numbers (e.g., 3.2 × 10^(−9) nm for a nanometre).

Fractional exponents: roots

Fractional exponents express roots. A unit fraction b^(1/n) is the nth root of b:

So 4^(1/2) = √4 = 2, 8^(1/3) = ∛8 = 2, 16^(1/4) = ⁴√16 = 2. More generally, a rational exponent p/q combines power and root:

Example: 8^(2/3) = (∛8)^2 = 2^2 = 4. Both orderings — root first then power, or power first then root — give the same answer. Starting with the root keeps intermediate numbers smaller, which is usually easier.

Decimal exponents

Any decimal exponent can be interpreted as a rational exponent or evaluated numerically. For instance, 2^0.75 = 2^(3/4) = ⁴√(2³) = ⁴√8 ≈ 1.6818. Irrational exponents like 2^π or e^√2 are defined through the limit of rational approximations and are always evaluated numerically.

Negative base with fractional exponents

This case requires care. For a negative base b < 0:

• Even-root fractions (½, ¼, ⅙, …): the result is complex (involves imaginary numbers). Example: (−4)^0.5 = 2i, not a real number.
• Odd-root fractions (⅓, ⅕, …): the result is real. (−8)^(1/3) = −2 because (−2)^3 = −8. The sign follows the base when the numerator is odd.

In most calculators and programming languages, Math.pow(−8, 1/3) returns NaN because the general floating-point rule flags all negative-base fractional powers as complex. This calculator detects odd-denominator cases and returns the correct real result.

The laws of exponents

Scientific notation and exponents

Scientific notation expresses numbers as m × 10^n where 1 ≤ |m| < 10. Powers of 10 are the backbone: each step up the exponent multiplies by 10. Some landmark values:

• 10^9 = 1 billion (a second of time on a fast CPU)
• 10^12 = 1 trillion (a terabyte in bytes)
• 10^23 ≈ Avogadro's number (atoms in a mole)
• 10^80 ≈ estimated number of atoms in the observable universe

The exponential function eˣ (where e ≈ 2.718) is the most important single function in mathematics. Every exponential growth or decay process — compound interest, radioactive decay, population growth, charging capacitors — involves it.

Exponents in algebra

The rules for exponents are foundational to algebraic manipulation. Key techniques:

• Simplifying expressions: Combine terms with the same base using the product and quotient rules before evaluating numerically.
• Solving exponential equations: If 2^x = 32, rewrite 32 as 2^5 to read off x = 5. When bases differ, use logarithms: x = log_b(n).
• Binomial expansion: (a + b)^n expands using the Binomial Theorem; exponent rules govern each term.

Common mistakes

• −2² ≠ (−2)²: By standard order of operations, −2² = −4 (negate after squaring), but (−2)² = 4 (square the negative number). Always parenthesise a negative base.
• aᵐ × bⁿ ≠ (ab)^(m+n): The product rule only applies when the bases are the same.
• (a + b)² ≠ a² + b²: This is one of the most common algebra errors. The correct expansion is a² + 2ab + b².