Exponent Calculator

An exponent (also called a power) indicates repeated multiplication. In 2^5, the base is 2 and the exponent is 5, meaning 2×2×2×2×2 = 32. Exponents are written as superscripts: 2&sup5; or as 2^5 in text notation. The result is called a power. Exponentiation is the mathematical operation of raising a base to a power.

Any non-zero number raised to the power of 0 equals 1. This is a mathematical definition, not derived from repeated multiplication. 5^0 = 1. 1000^0 = 1. (−7)^0 = 1. The expression 0^0 is considered indeterminate and is typically left undefined, though in some contexts (combinatorics, set theory) it is taken to equal 1 for convenience.

A negative exponent means take the reciprocal. X^(−n) = 1 / X^n. Examples: 2^(−1) = 1/2 = 0.5. 2^(−3) = 1/8 = 0.125. 10^(−2) = 1/100 = 0.01. Negative exponents are used in scientific notation for very small numbers: 3 × 10^−9 = 3 nanometers = 0.000000003.

Fractional exponents represent roots. X^(1/2) = square root of X. X^(1/3) = cube root of X. X^(m/n) = nth root of X raised to the mth power. Example: 8^(2/3) = (cube root of 8)^2 = 2^2 = 4. This connects exponents and roots: x^(1/n) = ⁿ√x. So 9^0.5 = 9^(1/2) = √9 = 3.

The main laws: Product rule: X^a × X^b = X^(a+b). Quotient rule: X^a / X^b = X^(a−b). Power rule: (X^a)^b = X^(ab). Zero exponent: X^0 = 1. Negative exponent: X^(−n) = 1/X^n. Product base: (XY)^n = X^n × Y^n. These rules apply when bases are the same (for product and quotient rules). They allow complex exponential expressions to be simplified.

Exponents grow incredibly fast. 2^10 = 1,024. 2^20 ≈ 1 million. 2^30 ≈ 1 billion. 2^32 = 4,294,967,296 (the 32-bit integer limit). 2^64 ≈ 1.8 × 10^19. 2^100 ≈ 1.27 × 10^30. A googol = 10^100. This explosive growth is why compound interest, viral spread, and computer storage capacity follow exponential patterns.

2^32 = 4,294,967,296 (about 4.3 billion). This is the maximum value a 32-bit unsigned integer can store, which is why old computers and software were limited to about 4 GB of RAM (2^32 bytes). Similarly, 2^64 ≈ 18.4 quintillion, which is the limit of 64-bit systems. IPv4 has 2^32 addresses; IPv6 has 2^128 addresses.

Strictly: in 2^5, the "exponent" is 5 (the number that indicates how many times to multiply). The "power" is the result: 32. However, in everyday use, these terms are often used interchangeably. "Raise 2 to the power of 5" and "2 to the exponent 5" mean the same thing. "Power of 2" typically means a number of the form 2^n: 1, 2, 4, 8, 16, 32, 64...

Scientific notation expresses numbers as a×10^b, where 1 ≤ |a| < 10. Examples: 299,792,458 (speed of light in m/s) = 2.998 × 10^8. 0.000000001 (1 nanometer) = 1 × 10^−9. The exponent of 10 shows how many places to move the decimal point: positive exponent = large number, negative exponent = small number. It makes very large and very small numbers manageable.

0^0 is mathematically indeterminate (no universally agreed value) because two rules conflict: any number to the 0 = 1, but 0 to any positive power = 0. In practice: in combinatorics and set theory, 0^0 = 1 is used (for example, the number of functions from an empty set to an empty set is 1). In calculus, limits involving 0^0 depend on how you approach it, giving different answers. Calculators often return 1 or an error for this case.