How to convert normal number to a number with power.

Question

I wanted to ask all of you how can I convert a normal number to a number with a power. It does not have to be the power of $2$ but it must be a power. For example :

$16$ convert to $ 4^{2}$

$4$ convert to $ 2^{2} $

Thanks.

Answer 1

To write a number as a power, you can try to factorise it into primes. For example, $$225=3\times 3\times 5\times 5=3\times 5\times 3\times 5=(3\times 5)^2=15^2$$ or $$216=2\times 2\times 2\times 3\times 3\times 3=2\times 3\times 2\times 3\times 2\times 3=(2\times 3)^3=6^3.$$

Answer 2

A more general approach would be something like :

$$a=a^{1}=a^{\frac{n}{n}}={(a^{\frac{1}{n}})}^n.$$

Assuming the $n$-th root exists for $a$, you will get an integer raised to another integer this way. For example given $a=16$,

$16={(16^{\frac{1}{2}})}^2=4^2$ for $n=2$

$16={(16^{\frac{1}{4}})}^4=2^4$ for $n=4$.