I have a basic question, but, i don't really remember how to solve this kind of problem. I know the result must be $a = b$, but I don't know how to get there.
"Determine the relationship between a and b where a and b are natural numbers that express the digit numbers of $x = 4 ^ {12} * 5 ^ {20}$ and $y = 4 ^ {14} * 5 ^ {18}$ respectively."
Note $4=2^2 \Rightarrow 4^n=2^{2n}$ and $2^m \cdot 5^m = (2\cdot 5)^m=10^m$
So $$x=4^{12}\cdot 5^{20}=2^{24} \cdot 5^{20}$$ $$ =2^{4+20}\cdot 5^{20}=2^{4} \cdot 2^{20}\cdot 5^{20}$$ $$ =2^{4} \cdot 10^{20}=16\cdot 10^{20}$$
Thus $x$ has $2+20=22$ digits.
Similarly show $$y=2^{10}\cdot10^{18}=1024\cdot10^{18}$$
Then $y$ has $4+18=22$ digits.