What kind is an algorithm which "opens" large natural base numbers with natural number exponents?

Manually 3^5 is easy as it only demands 5 repetitive multiplications (3*3*3*3*3) in order to find precise answer (=243).

But 3^4567 does not fit in screen of a typical calculator.

Even if we use computers, they usually give us approximate values in the form of base number 10 exponents (like 1,56789 * 10^234; this is just an example what we can see in calculator or computer screen, not an real approximation of 3^4567).

So, what I am looking for is an algorithm that gives precise values for these stupendously large numbers, not an approximative method.

Is there any shorcut method, or does our computers actually just use raw power and repetitively multiply 3 4567 times in this example? As far as I know, Taylor series can be used in order to find precise enough decimal values for logarithms to be found, so that they can be used to "open", say, 3^4567 with the required precision...

But does that calculation of appropriate Taylor series offers any advantage compared to raw repetitive multiplications?

I do not know, but it is my guess that it does not.

I am aware, and I apologize, that my question may sound like irritating to many MHF readers, but I have a certain application in my mind which requires answer to this question.