exponent

An exponent is a mathematical notation that indicates how many times a number, known as the base, is multiplied by itself. It is a shorthand way of expressing repeated multiplication. For example, in the expression 3^4, the number 3 is the base, and 4 is the exponent, meaning 3 is multiplied by itself four times: 3 × 3 × 3 × 3, which equals 81.

The exponent is also called the power or index. It can be a positive integer, zero, or a negative number. When the exponent is a positive integer, it signifies repeated multiplication of the base. When it is zero, the result is always 1, provided the base is not zero (by convention, 0^0 is undefined in many contexts). A negative exponent indicates the reciprocal of the base raised to the corresponding positive exponent; for example, 2^−3 = 1/(2^3) = 1/8.

Exponents have several important properties. The product rule states that a^m × a^n = a^(m+n), where the bases are the same. The power rule states that (a^m)^n = a^(m×n). The quotient rule states that a^m / a^n = a^(m−n), assuming the base is non-zero. These rules facilitate the simplification of exponential expressions and are fundamental in algebra, calculus, and various scientific disciplines.

Exponents are also used in scientific notation to express very large or very small numbers compactly, such as 3.0 × 10^8 for the speed of light. Overall, exponents are a fundamental concept in mathematics, underpinning many methods of quantitative analysis and problem-solving.