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Exponent

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An exponent is one of those tiny numbers at the upper right that means you have to multiply something that number of times: for example, $53$ means 5 multiplied by itself 3 times or $5 \times 5 \times 5;$ the number 3 is called the exponent.

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.