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Given composition xoy=2(x-3)(y-3)+3, solve 5^xo5^x=11?

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ruals | (Level 1) Salutatorian

Posted July 3, 2013 at 4:37 PM via web

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Given composition xoy=2(x-3)(y-3)+3, solve 5^xo5^x=11?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted July 3, 2013 at 5:05 PM (Answer #1)

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You need to replace `5^x` for `x` and `y` in the given law of composition, such that:

`5^xo5^x = 2(5^x - 3)(5^x - 3) + 3`

Replacing `2(5^x - 3)(5^x - 3) + 3` for `5^xo5^x` in equation, yields:

`2(5^x - 3)(5^x - 3) + 3 = 11 => 2(5^x - 3)(5^x - 3) = 11 - 3`

`2(5^x - 3)(5^x - 3) = 8 => (5^x - 3)(5^x - 3) = 4 => (5^x - 3)^2 = 4 => (5^x - 3)^2 - 4 = 0`

You need to convert the difference of squares into a product, such that:

`(5^x - 3 - 2)(5^x - 3 + 2) = 0 => (5^x - 5)(5^x - 1) = 0`

Using the zero product rule yields:

`{(5^x - 5 = 0),(5^x - 1 = 0):} => {(5^x = 5),(5^x = 1):}`

`{(5^x = 5^1),(5^x = 5^0):} => {(x = 1),(x = 0):}`

Hence, evaluating the solutions to the given equation, using the given law of composition, `5^xo5^x = 11` , yields `x = 0, x = 1.`

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