document.write( "Question 1131959: the 41st root of 2 199 023 255 552 is an integer. What integer, and how was the answer found? \n" ); document.write( "

Algebra.Com's Answer #748721 by greenestamps(13209) $\"\"$ $\"About$

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\n" ); document.write( "There are lots of things we can do using estimation and logical reasoning to determine that the integer is 2.

\n" ); document.write( "To begin with, the units digit is 2, so the integer has to be even. After that....

\n" ); document.write( "(1)10^41 is a number with 42 digits. If the 41st power of an integer is a number with only 13 digits, then the integer has to be VERY small. So just with that much reasoning, the integer almost has to be 2.

\n" ); document.write( "(2) We can gain confidence that the integer is 2 by noting that the units digit of the given number is 2. No power of either 4 or 6 has units digit 2, so the only possibilities for the integer are 2 and 8. And 8 is almost certainly too big.

\n" ); document.write( "(3) Finally, to confirm that 2 is the integer we are looking for, we can use the (very useful!) approximation 2^10 = 10^3 to see that 2^41 will be a number with about 13 digits:

\n" ); document.write( "2^41 = 2(2^40) = 2((2^10)^4) = 2((10^3)^4) = 2(10^12)

\n" ); document.write( "That is a 13-digit number with first digit 2 -- which is what the given number is. \n" ); document.write( "

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