How to divide exponents with different base numbers

Question

Could not find a calculator online that could handle my large number. Could some help me with the solution for this very large number, I've forgotten how to divide exponentials with different bases.

$$\dfrac{2^{64^{64}}}{10^{80}}=?$$

Please show how you did it so I can do it myself too.

Answer 1

Hint: $10 = 2\cdot 5$ so $10^{80} = 2^{80} \cdot 5^{80}$.

Answer 2

let $$x=\frac{2^{{64}^{64}}}{10^{80}}$$ then you will get $$\ln(x)=64^{64}\ln(2)-80\ln(10)$$

Answer 3

Dr. Graubner's answer is spot on but for a different approach that might be easier to use when you don't have a calculator (or are fairly bad with logs) you can take the following very rough approach.

$10^{80}<16^{80}=2^{4*80}=2^{320}$ so $$\frac{2^{64^{64}}}{10^{80}}>\frac{2^{64^{64}}}{2^{320}}=2^{64^{64}-320}$$

Now $64^{64}-320$ is so close to $64^{64}$ it's almost not worth bounding but if you really wanted to (and the error here is stupid large) you can notice that $64^{63}>320$ so $64^{63}*64-320>64^{63}*64-64^{63}=64^{63}*63>64^{63}$

Altogether then you get

$$\frac{2^{64^{64}}}{10^{80}}>2^{64^{63}}$$

The error is REALLY big but at least you get some idea without having to do any computations.

EDIT: I let Wolfram Alpha actually do the computations for me and the error you get my way is stupidly big.

$$2^{64^{64}}\approx 10^{10^{115.0741281073144}}$$ and ... wait for it $$\frac{2^{64^{64}}}{10^{80}}\approx 10^{10^{115.0741281073144}}$$

whereas $$2^{64^{63}}\approx 10^{10^{113.2679481333305}}$$

Further EDIT:

Thinking about this really should not have surprised me. Dividing by $10^{80}$ means throwing away $80$ zeros. The top has roughly $10^{115}$ zeros so ofcourse you can't possibly see the difference.