# Fractional Power Interpretation

I have a following query in my mind. It has been in my mind since i was a kid.

I know that 2^3 means that multiply 2 three times,3^-2 means multiply (1/3) two times.What does 2^(0.22) means. multiply 2 how many times, i mean what is logic behind fractional power?

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Duplicate of math.stackexchange.com/questions/132703/… – Dan Feb 18 '14 at 7:11
No duplication please.He was asking for how to calculate i want explanation.I am not asking how i am asking why? so please take back this duplicate tag on me. – Naseer Ahmed Feb 18 '14 at 7:14
Since exponentiation is nothing more than repeated multiplication, it is fairly easy to show that $(a^b)^c=a^{bc}$ . Now notice that, by definition, $(\sqrt[n]a)^n=a$ . Now just compare the two identities. – Lucian Feb 18 '14 at 7:36

Since $a^{p/q}$ is defined by $\sqrt[q]{a^p}$,we have $2^{0.22}=\sqrt[50]{2^{11}}$. The logic behind this definition is to define fractional exponents in a way such that all power rules are still valid.

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That is quite an interesting question, most people never ask themselves what that means...

The answer by Michael Hoppe is correct, but consider for example $2^\pi$, what does it mean ? Now we can't use Michael's method because we can't write $\pi$ as a rational number, that's we can't write $\pi = \frac{p}{q}$, now what ?

There is quite an interesting solution to this, as

$$\pi = 3.14159265359...$$

we can write, for example, $$\pi \approx \frac{314159}{100000}$$

that is a very good approximation to $\pi$, and now we can use Michael's method!

Ok, that is an "intuitive" answer, but the fact is that we define $2^\pi$ as the limit of the sequence $2^{\frac{p}{q}}$ as $\frac{p}{q}$ approaches $\pi$.

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