exponentform

Exponent form, or exponential form, expresses numbers as powers of a base. A real number a raised to an exponent b is written as a^b. The base a is typically a positive real number, and b can be an integer, a fraction, or any real number in many contexts. For integer exponents, a^n means multiplying a by itself n times. Negative exponents denote reciprocals: a^−n = 1/(a^n). The zero exponent rule states a^0 = 1 for a ≠ 0.

Basic rules of exponent form include: a^m · a^n = a^(m+n); (a^m)^n = a^(mn); and (a^m)/(a^n) = a^(m−n). When multiplying or dividing numbers with the same base, these rules allow simplification. For expressions with different bases, equivalent forms or factoring are often used.

Fractional exponents express roots: a^(p/q) equals the qth root of a^p, provided a is nonnegative when q is even (to stay in the real numbers). In general, fractional and irrational exponents can extend to real or complex values, with attention to domain restrictions.

Scientific notation is a common application of exponential form in science and engineering. It writes numbers as m × 10^n, where m is a decimal between 1 and 10 (or −1 and 0, depending on conventions) and n is an integer. This is a specialized exponential form designed for convenience with very large or very small quantities.

Examples: 16 = 4^2 = 2^4; 2^−3 = 1/8; 9^(1/2) = 3; 7^0 = 1. Exponent form is a foundational tool in algebra, calculus, and applied sciences for compactly representing and manipulating powers and growth rates.

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