SOLUTION: 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 possible value of n among these pai

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
possible value of n among these pairs?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
Of all the pairs (x,n) whose product is the same number, the greatest will be paired with the least positive .
Since , the greatest integer that could be is
So we have to find =(25)*(4^4)*(8^(8/3))*(16^(3/4))
Either way, that expression is hard to type.
Luckily, it can be simplified, because
--> 16^(3/4)=
--> 8^(8/3)=
and, while we are at it,
-->
So substituting all that, we get a large number that can be calculated with a calculator,
or with pencil and paper:

NOTE 1:
If it was specified that x, and n could not be 1,
then the least possible positive x would be 2,
and that would make the greatest possible n.

NOTE 2:
Since the positive factors are all of the form
with being any of the 20 integers from 0 to 19 including 0 and 19,
and being 0, 1, or 2 (just 3 choices).
That makes the total number of positive factors .
That makes 60 (x,n) ordered pairs, or 30 {x,n} sets of factors (if order does not matter).
Considering the negative factors doubles the possibilities.
RELATED QUESTIONS
(8 to the-3 power) * ( 4 to the 4th... (answered by josgarithmetic)

what is the slimplest form of x+x to the second power+ x to the 3 power+ x to the 4... (answered by Fombitz)

what is n to the 4th power divided by n to the 8 power? (answered by KMST)

8 to power of 6 divided by 4 to the power of 3 x 2 to the power of... (answered by jsmallt9)

Solve for X. 8raised to the power of x+4 is equal to 16raised to the power of 4x-3.... (answered by josgarithmetic,ikleyn,josmiceli)

What is (3/4) to the 4th power equal? (answered by jim_thompson5910)

how are the expressions 3 to the 4th power and 4 to the 3rd power... (answered by checkley71)

what is 8 to the power of 2 x 8 to the power of 3 = 8 ? (answered by Tatiana_Stebko)

can you solve these problems in steps? Set2 1. solve the equation 5x+3=-15, for x... (answered by rapaljer)