An exponent tells you how many times to multiply a number by itself. In 2⁵, the base is 2 and the exponent is 5, meaning 2 × 2 × 2 × 2 × 2 = 32. This calculator handles positive exponents, but also the two cases that most commonly confuse people: negative exponents and fractional exponents.
A negative exponent means "take the reciprocal," not "make the result negative" — a very common misreading. 2⁻³ is not −8; it's 1/(2³) = 1/8. The negative sign in the exponent flips the base to a fraction; it doesn't flip the sign of the final answer. Worked example: 5⁻² = 1/(5²) = 1/25 = 0.04.
A fractional exponent represents a root. x^(1/2) means the square root of x, and x^(1/3) means the cube root of x — the denominator of the fraction tells you which root to take. More generally, x^(a/b) means take the b-th root of x, then raise the result to the power of a (or equivalently, raise x to the power a first, then take the b-th root — both orders give the same answer). Worked example: 8^(2/3) means the cube root of 8 (which is 2), then squared: 2² = 4.
Any nonzero number raised to the power of 0 equals 1 — this includes surprising-looking cases like 1,000,000⁰ = 1 and even (−5)⁰ = 1. This isn't an arbitrary rule; it falls directly out of the pattern of dividing consecutive powers of the same base: 2³ ÷ 2² = 2¹ = 2, 2² ÷ 2¹ = 2⁰... which only stays consistent if 2⁰ = 1.
The exponent rules that make combining powers manageable: multiplying same-base powers adds the exponents (2³ × 2⁴ = 2⁷), dividing same-base powers subtracts them (2⁷ ÷ 2³ = 2⁴), and raising a power to another power multiplies the exponents ((2³)² = 2⁶). These rules only apply directly when the base is the same on both sides — 2³ × 3² cannot be combined into a single power using these shortcuts, because 2 and 3 are different bases.
Exponents scale extremely fast, which is worth feeling out with a concrete comparison: 2¹⁰ = 1,024, but 2²⁰ isn't merely double that — it's 1,024², which is 1,048,576, over a thousand times larger from doubling the exponent, not the base. That's the whole idea behind "exponential growth" as a description of anything that compounds — population growth, compound interest, or viral spread all follow this same pattern where a fixed percentage increase per period produces an accelerating, not linear, total over time.
A very common mistake is treating exponentiation like multiplication when it comes to distributing over addition: (a + b)² does not equal a² + b² (it actually expands to a² + 2ab + b², via the binomial expansion) — this is one of the most frequently repeated algebra errors, sometimes called the "freshman's dream" for how tempting the incorrect shortcut looks. Another common mistake is misreading order of operations: −3² is not the same as (−3)², because exponentiation applies to the 3 alone before the negative sign, giving −9, whereas (−3)² with explicit parentheses gives 9. Powers of 2 specifically are worth some familiarity beyond pure math class, since they underpin how computers represent and store data: a byte holds 2⁸ = 256 distinct values, and storage capacities (kilobytes, megabytes, gigabytes) are traditionally built on powers of 2 close to, but not exactly matching, their decimal-sounding names (1 kilobyte in the strict binary sense is 2¹⁰ = 1,024 bytes, not exactly 1,000). This is also directly connected to this site's binary-numeral pages — the binary representation of any number is, at its core, an expression of that number as a sum of powers of 2.
For a fuller treatment of these rules with more worked examples, see Understanding Exponents and Powers.