`(5^(2x)-2^(2x)) / (5^x-2^x)` Please simplify and give your answer in positive indices

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You may use exponential laws, hence, you may write `5^(2x)` as `(5^x)^2` and you can do the same with the power ` 2^(2x)` , such that:

`(5^(2x) - 2^(2x))/(5^x - 2^x) =((5^x)^2 - (2^x)^2)/(5^x - 2^x)`

You may convert the difference of squares `(5^x)^2 - (2^x)^2` into a product, such that:

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

Replacing `(5^x - 2^x)(5^x + 2^x)` for `(5^x)^2 - (2^x)^2` , yields:

`((5^x - 2^x)(5^x + 2^x))/(5^x - 2^x)`

Reducing duplicate factors yields:

`(5^(2x) - 2^(2x))/(5^x - 2^x) = 5^x + 2^x`

Hence, performing the required simplifications yields `(5^(2x) - 2^(2x))/(5^x - 2^x) = 5^x + 2^x.`

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