# Non-Integer, Rational Exponents

How do we interpret exponential terms where the exponent is not an integer but a *non-integer rational number*? For example, consider the exponential term \({3^{\left( {\frac{1}{2}} \right)}}\). What is the meaning of this term? This is a term which when multiplied with itself should give us 3. In simpler terms, this is the square root of 3. But what about \({3^{\left( {\frac{1}{5}} \right)}}\) ? This is a term which when multiplied with itself 5 times, should give us 3. In other words, this is the fifth root of 3. Thus, we see that any exponential term of the form \({b^{\left({\frac{1}{n}}\right)}}\) where *n* is an integer, has a simple meaning: this is the *n*^{th} root of *b*, a number which when multiplied *n* times with itself will generate *b*.

What if *n* is a negative integer? Once again, we take the reciprocal, and then the *n*^{th} root. For example, consider the exponential term \({\pi ^{\left( { - \frac{1}{7}} \right)}}\). This can be written as\(\frac{1}{{{\pi ^{\left( {\frac{1}{7}} \right)}}}}\). The interpretation of this term will be: it is the reciprocal of the seventh root of \(\pi \).

Now, consider the exponent to be a general rational number, of the form\(\frac{p}{q}\). What interpretation do we attach to the exponential term \({b^{\left( {\frac{p}{q}} \right)}}\)? We can rewrite this in two ways (can you see how?):

\[{\left( {{b^p}} \right)^{\left( {\frac{1}{q}} \right)}}\;\;{\rm{or}}\;\;{\left( {{b^{\left( {\frac{1}{q}} \right)}}} \right)^p}\]

Thus, the interpretation will be: this is the *q*^{th} root of (*b* raised to the power *p*), or equivalently, the *p*^{th} power of (the *q*^{th} root of *b*). Note that both interpretations imply one and the same thing. For example,

\[{7^{\left( {\frac{{12}}{{23}}} \right)}} = {\left( {{7^{12}}} \right)^{\frac{1}{{23}}}} = {\left( {{7^{\frac{1}{{23}}}}} \right)^{12}}\]

can be thought of as the 23^{rd} root of (7 raised to the power 12), or the 12^{th} power of (the 23^{rd} root of 7).

We are now in a position to interpret the meaning of any exponential term where the exponent is an arbitrary rational number. But what meaning do we assign to an exponential term where the *exponent is an irrational number*?