how to calculate in detail ,4984209207 is the fifth power of what number,without calculator?

Answer 1

You asked the question right this time. :) The answer is 87. See my notes in your first question.

OK, I see you added the wrinkle that you want to do this without a calculator. I'd say find the logarithm of that number, which is between 9 and 10, then divide the logarithm by 5 (since you want the 5th root). Use the logarithm table again to figure out about what the new number should be. You should get 80 to 90. Finally, use the fact that you know the number has to end in 7, giving you 87 as the answer.

Answer 2

I will assume you already know that the answer is an integer. Your number has 10 digits, and you know that 100 to the 5th power will have 10 zeros and therefore 11 digits. That gives you an upper estimate on the number you are looking for. On the other hand the number you have is very close to being 11 digit so you expect to see a number close to 100 rather than, say 50. Now the next clue comes from the last digit which is an odd number. So its 5th root can not be even. That leaves 1,3,5,7,9 as the only possibilities for the last digit of its 5th root. However it is also easy to rule out 1 and 5 right away as any power of a number ending in those will end in 1 or 5 again. Now this reduces your search to the numbers which have their last digits equal to 3, 7 and 9. Now using modular arithmetic check that 5th power of a number which end in 3 will be 3, which end in 7 will be 7 and which end in 9 will be 9. So here are your possibilities: 97, 87, 77,... Now it is a matter of trying a couple of numbers to see that the answer must be 87.

Answer 3

Dealing with a big number without a calculator is a pain.
Here, what I did i some kind of approximation and trial and error!

I know that 9^5 = 59049
so 90^5 = 5904900000

so the answer has to be less than 90

but it is definitely less than 90

from here, you could do trial and error
dealing with 89, 88....
?=89 * 89 * 89 * 89 * 89
?=89^2 * 89^2 * 89


solving this by my own gives me 87 as the answer..

there might be a better way of solving this
besides a calculators help.. hahaha

Answer 4

87.

How did I know? (besides reading the other answer).

Since the number has 10 digits, it must be a 2 digit number. Each time you add 1 to the power, the number must add 2 digits.

Since it ends in a '7', the units (1s) digit must be a 7 (7, 49, 343, 2401, 16807).

Since the result is approx 1/2 of 100^5, it must be near 90. (9, 81, 729, 6561, 59049). Since it's just below 5904900000 it must be just below 90, so the answer must be .87.