SOLUTION: If a,b,and c are positive integers, find the sum a+b+c if {{{ a^3bc = 7776 }}} and {{{ ab^3c=17496 }}} and {{{ abc^3=3456 }}}

Algebra ->  Equations -> SOLUTION: If a,b,and c are positive integers, find the sum a+b+c if {{{ a^3bc = 7776 }}} and {{{ ab^3c=17496 }}} and {{{ abc^3=3456 }}}      Log On


   

Question 1188519: If a,b,and c are positive integers, find the sum a+b+c if +a%5E3bc+=+7776+ and +ab%5E3c=17496+ and +abc%5E3=3456+
Answer by Shin123(626) About Me  (Show Source):
You can put this solution on YOUR website!
Multiplying all 3 equations together, you get a%5E5b%5E5c%5E5=7776%2A17496%2A3456.
If we take the prime factorizations of the numbers, we get
7776=2%5E5%2A3%5E5
17496=2%5E3%2A3%5E7
3456=2%5E7%2A3%5E3.
Multiplying that together, we get a%5E5b%5E5c%5E5=2%5E15%2A3%5E15.
Taking the 5th root, we get abc=2%5E3%2A3%5E3=216.
Taking the first equation and dividing the equation we just derived, we get a%5E2=36.
Doing the same for the other two, we get b%5E2=81 and c%5E2=16.
Since a,b,and c are positive integers, we have that a=6,b=9, and c=4.
Adding all of them up, we get highlight%28highlight_green%28highlight%28a%2Bb%2Bc=19%29%29%29.