What exactly goes raising a number to a fraction mean?

Question

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?

Answer 1

$$2^{\frac{2}{5}} = 2^{2\times \frac{1}{5}} = \sqrt[5]{2^{2}}$$

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. $$x^{\frac{p}{q}} = \sqrt[q]{x^p}$$

Answer 2

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.

Answer 3

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).

Answer 4

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.