Lesson WHY fractional exponents are defined the way they are

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A you recall, mathematics progressed from just knowing positive integers 1, 2, 3, ... , to understanding the concept of zero, to negative numbers, then to fractions and then to real numbers.

The same thing happened when mathematicians explored the concept of powers. First, they defined a natural power, like {{{2^3=2*2*2}}}. After mathematicians <A HREF=why-negative-exponents.lesson>figured out how to define negative exponents</A>, their attention focused on how to use fractional powers. This lesson discussed why did they define them as they did.

They wanted fractional powers to fit within the properties of powers, such as {{{a^x*a^y = a^(x+y)}}} (first property), and {{{(a^x)^y = a^(x*y)}}} (second property).

Consider, first, a simple case of a number to the power of 1 over a natural number:

We want the second property to help us. We'll use a clever trick and take number {{{a^(1/n)}}} to the degree of n:

{{{(a^(1/n) )^n}}}

Since we want the second property to hold, we can expand this:

{{{(a^(1/n) )^n = a^((1/n)*n) = a^1 = a}}}

{{{highlight( (a^(1/n) ) )^n = a}}}

By definition, the highlighted number that, when taken to nth degree, is equal to a, is {{{root( n, a) }}}.

So, we now know that {{{a^(1/n) = root( n, a ) }}}.

That's how mathematicians decided to define exponents with power in form {{{1/n}}}

Definition: {{{a^(1/n) = root( n, a ) }}}