Non-Integer, Rational Exponents

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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 nth 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 nth 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 qth root of (b raised to the power p), or equivalently, the pth power of (the qth 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 23rd root of (7 raised to the power 12), or the 12th power of (the 23rd 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?

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