how to simplify 2^(2a)-2^(a+2)

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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The expression 2^(2a)-2^(a+2) has to be simplified.

2^(2a)-2^(a+2)

=> (2^a)^2 - 2^2*2^a

=> 2^a(2^a - 4)

The expression 2^(2a)-2^(a+2) cannot be simplified though it is possible to factorize it as 2^a(2^a - 4)

sciencesolve's profile pic

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to write the factored form of the given expression, hence you should use the exponential laws such that:

`2^(2a)  = 2^(a+a) = 2^a*2^a`

`2^(a+2) = 2^a*2^2 = 4*2^a`

You need to substitute `2^a*2^a`  for `2^(2a)`  and `2^a*2^2`   for `2^(a+2)`  such that:

`2^a*2^a-4*2^a`

Notice that you may factor out `2^a`  such that:

`2^a*2^a- 4*2^a = 2^a(2^a - 2^2)`

Notice, if `a = 2n` , then `(2^a - 2^2) = (2^(2n) - 2^2) = (2^n - 2)(2^n + 2)` `2^a*2^a- 4*2^a = 2^a(2^n - 2)(2^n + 2)`

Hence, factorizing the given expression yields `2^(2a) - 2^(a+2) = 2^a(2^a - 2^2).`

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