Elliot Temple posed the following math puzzle: How many square roots does it take to get 10^{100100} down to a small number? (And what number is it?)
Note that a^{bc} = a^{(bc)}.
Answer: 665 square roots takes you down from 10^{100100} to around 4.5.
Proof:
Here, lg means the base 10 logarithm, and dot (·) means multiplication. I will use the following facts:
- (a^{b})^{c} = a^{bc}
- x^{a}x^{b} = x^{a+b}
- x = 10^{lg x}
Lemma:
a^{bc} = 10^{10c lg b + lg lg a}
Example: 2^{23} = 2^{8} = 256 = 10^{103 lg 2 + lg lg 2}
Proof:
a^{bc} | |
= | (10^{lg a})^{(10lg b)c} |
= | (10^{lg a})^{10c lg b} |
= | 10^{lg a · 10c lg b } |
= | 10^{10lg lg a · 10c lg b} |
= | 10^{10c lg b + lg lg a} ▢ |
We can now use the lemma to show that 4.5^{2665} = 10^{10665 lg 2 + lg lg 4.5}. And 665 lg 2 + lg lg 4.5 ≈ 200.▢
You can also check the answer with Robert Munafo's Hypercalc, an online calculator that works with very large integers. To do this, enter 4.5^(2^665) in Hypercalc. It evaluates it to something very close to 10^10^200.
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