How to add exponents so that they are in the form $a^{b^{a}}$

Question

I have these exponents which I wish to add together

$$10^{10^{200}} + 10^{10^{100}}$$

How do I add them such that the resulting exponent is in the form $a^{b^{a}}$ where $a$ and $b$ are positive integers?

Answer 1

I don't think this is possible. Note that $${10}^{10^{200}} ={10}^{10^{100\times 2}} ={10}^{(10^{100})^2} ={10}^{(10^{100})(10^{100})} =({10}^{10^{100}}) {}^{(10^{100})}. $$ So, knowing that $x^m+x=x(x^{m-1} +1)$, we get $${10}^{10^{200}} +{10}^{10^{100}} =({10}^{10^{100}}) {}^{(10^{100})} +{10}^{10^{100}} =({10}^{10^{100}}) \Big( {10}^{10^{100} -1} +1\Big). $$

Answer 2

Note that

$$\large{10^{10^{200}}+10^{10^{100}}=10^{\overbrace{1000...000}^{\text{200 zeros}}}}+10^{\overbrace{1000...000}^{\text{100 zeros}}}\approx \large{10^{10^{200}}}$$