Very easy question: How Do I understand the language of basic Mathematics operations? Yes. So Even though I understand Calculus and I'm Undergraduate Physics student, I can't wrap my head around simple mathematics conundrums, 
like x=something, basically making one number equal to something else, which by this logic is the same thing as the number in question. e.g:
$$ 27= \sqrt[4]{531441} $$ - so here I don't understand how does one's mind come up with it? Sure, I understand that $$ 531441=27*27*27*27 $$
But what does one's mind need to have programmed inside of it to have this magical number/ math operations skills to be so obvious and instant? - I think here about instantanenous ability to transform simple numbers in to these weird expressions, which sure are logical, but SO much not INTUITIVE and Over the top???
Have a look: 
$$ 6= \log_5(15625)\ $$ 
sure, If I were to think about it I could've eventually come up with derivation of this, using some basic logarithm properties, but how come I do these operations at an instant inside of my head, without much of thinking, if at all?
 A: Are you referring to some real-life situations where you've been stumped by people's ability to quickly do calculations in their heads? (Not that this has much to do with mathematics as such - calculators can do it even faster!) I can assure you that, at the most basic level, people who either like the mathematics as a subject, or have had a lot of experience with calculation, probably have remembered a lot of (sort-of curious) results. In other words, for some people, it is just joy (rather than a chore) to remember and play with various numbers.
For example, powers of two ($1, 2, 4, 8, 16, ...$ and maybe even up to $2^{64}=18,446,744,073,709,551,616$ because of curious stories such as https://en.wikipedia.org/wiki/Wheat_and_chessboard_problem), powers of three ($1, 3, 9, 27, 81$ etc.) or of five ($1, 5, 25, 125, 625$ etc.), digits of $\pi$ ($3.14159265358979323846...$) etc. 
With your example, people may recognise that $15625=5^6$ is one of powers of $5$, and that $531441=3^{12}$ is one of powers of $3$ (so $\sqrt[4]{531441}=\sqrt[4]{3^{12}}=3^{12/4}=3^3=27$ - another power of $3$).
A: Let's deal with the first example of finding $\sqrt[4]{531{,}441}$. Few people know that $27^4 = 531{,}441$, so the rest of us usually resort to divisibility rules. noting that the sum of the digits of $531{,}441$ is a multiple of nine [which implies that $531{,}441$ is a multiple of nine as well], we have $513{,}411 = 9\times 59{,}049$. Similarly noting that the sum of the digits of $59{,}049$ is also a multiple of nine, we have $531{,}441 = 9\times9\times6561$. Continuing this process we eventually end up with $$531{,}441 = 9^6 = (3^2)^6 = 3^{12} = (3^3)^4 = 27^4$$
The second example also uses divisibility rules, but with fives instead of nines. Using a similar method we eventually come up with $15{,}625 = 5^6$. But a leaner approach may be to split $15{,}625$ into $15{,}000 + 625$, and note that one number is a multiple of $5{,}000$ [which is a multiple of $5^4$ because $5{,}000= 5\cdot 1000 = 5 \cdot (2\cdot5)^3$], and the other is equal to $25^2 = 5^4$. So dividing we have $\frac{15{,}625}{625}=25$, which implies that $$15{,}625 = 625\cdot 25 = 25^3=5^6$$
Of course this happens mentally much faster than I can type, and with fewer words. If I were to vocalize my mental explanation to myself it would sound more like "da ding, da ding, da ding, cha-ching. Okay, got it."
One way at becoming skilled at mathematizing is to take some practice problems and not solve them in the same way all the time, but instead to develop multiple approaches. This way you will give yourself multiple strategies to choose from, which, over time, and with practice, you eventually earn a sense of which ones are most efficient in particular situations.
I'm saying all of this, noting that there are some quantities [as other commentators have already pointed out] that are just known and memorized ... numbers like $\sqrt 2, \pi, e, \phi$, and of course who could forget $625=25\times25$ (it even rhymes .... which begs the question, can a word rhyme with itself? ... but maybe I'll save that one for social media.).  
