Power of powers.

Calculus Level 3

How many digits are there in

$87^{13} + 13^{87} + 55^{45} + 45^{55} ?$

Hint: Notice that $87 +13 = 100, 55 + 45 = 100$ .

The answer is 97.

2 solutions

Samuel Li
Jan 1, 2015

Note that $13^{87}$ is 97 digits long, as can be calculated from $\lceil87*log(13)\rceil$ . This is larger than any of the other addends, by several orders of magnitude. So, you can effectively ignore the other terms, and the $13^{87}$ term dominates, making the whole number 97 digits long.

Brock Brown
Dec 28, 2014

Cheater's method:

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num = 87**13 + 13**87 + 55**45 + 45**55
answer = len(str(num))
print "Answer:", answer

Sorry for violating the spirit of the problem.

There's a trick.

Adam Phúc Nguyễn - 6 years, 5 months ago

And what is that trick?

Trevor Arashiro - 6 years, 5 months ago

Please tell us the trick so that we can tackle such type of problems. Thanks,

Prabir Chaudhuri - 6 years, 5 months ago

Power of powers.

The answer is 97.

2 solutions

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