I apologise for asking something so fundamental, but what exactly does $$2^\frac{2}{5}$$
actually mean? I get raising a whole number to another whole number $$x^y$$ means you are multiplying x with itself y times, but what does it mean when y is a fraction?
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$\begingroup$Intuitively, raising a number to a power like 2/5 can be thought of as first squaring the number, and then taking the fifth root of the number. Performing these operations in the opposite order also gives the same result.
$\endgroup$ $\begingroup$Note the following law for integers $ a^{x+y}=a^x\cdot a^y $. Now if we want to generalize this rule to fraction we note that $$a=a^{1}=a^{\frac{1}{2}+\frac{1}{2}}=a^{\frac{1}{2}}\cdot a^{\frac{1}{2}}$$ i.e. $a^{1/2}$ is the number which multiplies with it self becomes $a$ or in other words $a^{1/2}=\sqrt{a}$. In the same way $a^{1/b}$ should be the $b$:th root out of $a$.
Now for fractions we define $a^{\frac{b}{c}}= (a^{b})^{\frac{1}{c}}$ i.e. we take the $b$:th power of $a$ and then take the $c$:th root of that number. In your case $$2^{2/5}= (2^{2})^{1/5}=4^{1/5}$$ which is the $5$:th root of 4 (a number which is irrational and thus not nicely expressed in other ways).
$\endgroup$ $\begingroup$$$2^{\frac{2}{5}} = 2^{2\times \frac{1}{5}} = \sqrt[5]{2^{2}}$$
For any rational number $m = \frac{p}{q} \ | \ p,q \in \mathbb{N} $, $\ x^m$ is interpreted as the $q^{\text{th}}$ root of $x$ raised to the power of $p$ i.e. $$x^{\frac{p}{q}} = \sqrt[q]{x^p}$$
$\endgroup$ $\begingroup$Fractional powers work just like integer powers, and I'll show you how.
How many 5^1s do you have to multiply together to get 5^2? Two, because 2 is twice as big as 1. And how many 5^50s do you multiply to get 5^100? Two, because 100 is twice as big as 50.
So how many 5^(1/2)s do you multiply together to get 5^1? Two, because 1 is twice as big as 1/2. And what number, multiplied by itself, equals 5? The square root of 5.
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