Irrational Exponents

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What meaning to we assign to the exponential term $${2^{\sqrt 2 }}$$? since  $$\sqrt 2$$ cannot be written in the form $$\frac{p}{q}$$, we cannot apply the interpretation we used for rational exponents. In fact, some of you may ask, is this term even defined – does it even have a well-defined value? Yes, it does – understanding the interpretation of irrational exponents is important.

Let’s ask ourselves a slightly different question. Instead of asking – what is the meaning of this term, we ask ourselves: how do we calculate this term, if we had to? We know the approximate value of $$\sqrt 2$$: $$\sqrt 2 \approx 1.414$$. Thus, we could now say that the approximate value of our exponential term is:

${2^{\sqrt 2 }} \approx {2^{1.414}} = {2^{\left( {\frac{{1414}}{{1000}}} \right)}}$

We do know how to interpret (and calculate) the term on the right side above. We can raise 2 to the power 1414 and take the 1000th root of the resulting value.

But now you could object and say: this is just an approximation, and not the actual value of our exponential term. To that, one could reply: alright, then lets calculate it more accurately, by taking a better approximation of $$\sqrt 2$$: $$\sqrt 2 \approx 1.41421$$. And so,

${2^{\sqrt 2 }} \approx {2^{1.41421}} = {2^{\left( {\frac{{141421}}{{100000}}} \right)}}$

Once again, we know how to interpret and calculate the right hand side term. If you were to now say: this is still an approximation, then one could in turn say: we can take an even better approximation of $$\sqrt 2$$ and find the value of this term even more accurately. And we could keep doing this and come closer and closer to the actual value of this term.

To summarize: the exponential term 2 raised to the power$$\sqrt 2$$ is mathematically well-defined. If for some reason we had to calculate its value, we could take better and better rational approximations of $$\sqrt 2$$  for the exponent, and come closer and closer to the actual value of this exponential term. Generalizing this discussion, we see that we can now interpret exponential terms where the exponents are arbitrary irrational numbers.

Thus, in an exponential term, whether the exponent is an integer, a non-integer rational number, or an irrational number – we know how to interpret (and calculate) that term. This means that an exponent can be an arbitrary real number.

But what about the base itself? Can the base be any real number, or are there restrictions on the values the base can take?

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