Exponent

An exponent is a mathematical notation used to represent the operation of exponentiation. It is usually written as a superscript on a number or variable, called the base. For example, in the expression $5^{4}$, the base is 5 and the exponent is 4.

Exponents are typically numbers or variables themselves. The original usage for exponents is using a numerical base with a positive whole number exponent $n$ to represent the quantity obtained by multiplying the base by itself $n$ times. For example, the expression $5^{4}$ is defined to be 5 multiplied by itself 4 times: $5\times 5\times 5\times 5$.

Through centuries of development, the use of exponents has been extended to allow many other types of exponents, including negative integers, rational numbers, real numbers, complex numbers, and even matrices, sets, and other more complicated types of mathematical objects. These more exotic types of exponents no longer have meanings as simple as the product of a base with itself a certain number of times. Furthermore, in these more general contexts, not all exponential expressions have meaning (for instance, $0^{-1}$ is considered undefined). The large variety of meanings for exponential expressions and restrictions on when they have meaning at all often presents a strong barrier to mathematics students. Fortunately, the rules for exponentiation keep the same form and remain true regardless of the types of exponent being considered.

Extension of exponents to fractional and negative values

Originally, exponents were natural numbers: it's easy to see the meaning of an expression such as $10^{3}=10\times 10\times 10$. Rules for adding and multiplying exponents were noticed, and to extend the idea to fractions and negative numbers it was assumed that the same rules would apply. To define a meaning for a fractional value such as $10^{\frac {1}{2}}$, consider that, using a rule for multiplying exponents,

$(10^{\frac {1}{2}})^{2}=10^{{\frac {1}{2}}\times 2}=10^{1}=10$

Therefore $10^{\frac {1}{2}}$ must be ${\sqrt {10}}$. Values for many other numbers can be worked out similarly using cube roots and so on, and values for all real numbers can then be defined using limits.

To assign meaning to negative values of exponents, note the rule that

$b^{x}b^{y}=b^{x+y}.$

So, for example, to find the meaning of $10^{-3},$ consider

$10^{-3}\times 10^{4}=10^{-3+4}=10^{1}=10$

Therefore,

$10^{-3}={\frac {10}{10^{4}}}=0.001$

By a similar argument it can be established that $b^{0}=1$ for any base $b>1$.