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Question
Solve for x:
`(5^(x - 1) xx 25^(x + 2))/(5^x xx 125^(x - 2)) = 625`
Sum
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Solution
Rewrite everything in base 5:
25 = 52, 125 = 53, 625 = 54.
`(5^(x - 1) xx 25^(x + 2))/(5^x xx 125^(x - 2))`
= `(5^(x - 1) xx (5^2)^(x + 2))/(5^x xx (5^3)^(x - 2))`
= `(5^(x - 1) xx 5^(2x + 4))/(5^x xx 5^(3x - 6))`
= `5^(3x + 3)/(5^(4x - 6)`
= `5^((3x + 3) - (4x - 6))`
= `5^(9 - x)`
Set equal to 625 = 54:
`5^(9 - x) = 5^4`
⇒ `9 - x = 4`
⇒ `x = 5`
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