document.write( "Question 708707: There exist pairs of integers, x and n, for which xn = (25)(4 to the 4th power)(8 to the 8/3 power )(16 to the 3/4 power). What is the greatest
\n" ); document.write( "possible value of n among these pairs? \n" ); document.write( "

Algebra.Com's Answer #436210 by KMST(5328) $\"\"$ $\"About$

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Of all the pairs (x,n) whose product is the same $\"xn=N\"$ number, the greatest $\"n\"$ will be paired with the least positive $\"x\"$ .
\n" ); document.write( "Since $\"N=1%2AN\"$ , the greatest integer that could be $\"n\"$ is $\"N\"$
\n" ); document.write( "So we have to find $\"N=%2825%29%2A%284%5E4%29%2A%288%5E%288%2F3%29%29%2A%2816%5E%283%2F4%29%29%5Eor\"$ $\"N\"$ =(25)*(4^4)*(8^(8/3))*(16^(3/4))
\n" ); document.write( "Either way, that expression is hard to type.
\n" ); document.write( "Luckily, it can be simplified, because
\n" ); document.write( " $\"16=2%5E4\"$ --> 16^(3/4)= $\"%28root%284%2C16%29%29%5E3=%28root%284%2C2%5E4%29%29%5E3=2%5E3\"$
\n" ); document.write( " $\"8=2%5E3\"$ --> 8^(8/3)= $\"%28root%283%2C8%29%29%5E8=%28root%283%2C2%5E3%29%29%5E8=2%5E8\"$
\n" ); document.write( "and, while we are at it,
\n" ); document.write( " $\"4=2%5E2\"$ --> $\"4%5E4=%282%5E2%29%5E4=2%5E%282%2A4%29=2%5E8\"$
\n" ); document.write( "So substituting all that, we get a large number that can be calculated with a calculator,
\n" ); document.write( " $\"N=%2825%29%2A%282%5E8%29%2A%282%5E8%29%2A%282%5E3%29=highlight%2825%2A2%5E19%29\"$ or with pencil and paper:
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\n" ); document.write( "NOTE 1:
\n" ); document.write( "If it was specified that x, and n could not be 1,
\n" ); document.write( "then the least possible positive x would be 2,
\n" ); document.write( "and that would make $\"25%5E2%5E18\"$ the greatest possible n.
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\n" ); document.write( "NOTE 2:
\n" ); document.write( "Since $\"xn=25%2A2%5E19=2%5E19%2A5%5E2\"$ the positive factors are all of the form $\"2%5Ea%2A5%5Eb\"$
\n" ); document.write( "with $\"a\"$ being any of the 20 integers from 0 to 19 including 0 and 19,
\n" ); document.write( "and $\"b\"$ being 0, 1, or 2 (just 3 choices).
\n" ); document.write( "That makes the total number of positive factors $\"3%2A20=60\"$ .
\n" ); document.write( "That makes 60 (x,n) ordered pairs, or 30 {x,n} sets of factors (if order does not matter).
\n" ); document.write( "Considering the negative factors doubles the possibilities. \n" ); document.write( "

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