Exponents

Exponents, or INDICES. The product $5\; \backslash times\; 5\; \backslash times\; 5\; \backslash times\; 5$ is expressed thus, $5^4$, and the eighth power of $a$ thus, $a^8$. The numbers 4 and 8 are the exponents of these respective powers. In general, $a^n$ (where $a$ is any number or expression) stands for the $n$th power of $a$—i.e. the product of $n$ factors each = $a$, and there the value of this contracted notation is obvious. Introduced by Descartes, the theory of indices was speedily extended, and may now be said to affect algebraic operations of every conceivable kind. The two fundamental laws of indices are $a^m\; \backslash times\; a^n\; =\; a^\{m+n\}$, and $(a^m)^n\; =\; a^\{mn\}$. These, with certain necessary conventions, apply to all possible values of the exponents $m$ and $n$—integral or fractional, positive or negative, simple or complex, rational or surd, real or impossible, trigonometrical or logarithmic, &c. Minor results of the theory are such as these: $a^0\; =\; 1$, whatever $a$ may be; $a^\{-3\}\; =\; \backslash frac\{1\}\{a^3\}$, or $a^\{-n\}$ is the reciprocal of $a^n$; $a^\{\backslash frac\{1\}\{2\}\}\; =\; \backslash sqrt\{a\}$, $a^\{\backslash frac\{1\}\{3\}\}\; =\; \backslash sqrt[3]\{a\}$, $a^\{\backslash frac\{1\}\{4\}\}\; =\; \backslash sqrt[4]\{a\}$ = $(a^\{\backslash frac\{1\}\{2\}\})^\{\backslash frac\{1\}\{2\}\}$. An exponential equation is one in which the $x$ or $y$ occurs in the exponent of one or more terms, as $5^x\; =\; 800$. Its solution generally requires the use of logarithms. The exponential theorem gives a value of any number in terms of its natural logarithm, and from it can at once be derived a series determining the logarithm.