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Given 2^(x-y)=6 and 2^(x+y)=12, what is y?

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mozostax | Student, Undergraduate | (Level 2) eNoter

Posted August 9, 2013 at 3:36 PM via web

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Given 2^(x-y)=6 and 2^(x+y)=12, what is y?

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted August 9, 2013 at 3:54 PM (Answer #1)

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You need to use exponentiation rules, such that:

`2^(x - y) = 2^x/2^y`

`2^(x+y) = 2^x*2^y`

The problem provides the information that `2^(x - y) = 6` , hence, replacing `2^x/2^y` for `2^(x - y)` yields:

`2^x/2^y = 6`

The problem also provides the information that `2^(x + y) = 12` , hence, replacing `2^x*2^y` for `2^(x + y)` yields:

`2^x*2^y = 12 => 2^x*2^y = 6*2`

Replacing `2^x/2^y` for 6 yields:

`2^x*2^y = 2^x/2^y*2`

Reducing duplicate factors yields:

`2^y = 2/2^y => 2^y = 2^(1 - y)`

Equating the exponents, yields:

`y = 1 - y => 2y = 1 => y = 1/2`

Hence, evaluating y, under the given conditions and using the exponentiation rules, yields `y = 1/2` .

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