simplify: (2^a+3 - 4*2^a)/ (2^2a+1 - 4^a) I am not sure how to do this question. Could you please show each step. thanks :)

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llltkl | College Teacher | (Level 3) Valedictorian

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The given expression is `(2^(a+3) - 4*2^a)/ (2^(2a+1) - 4^a).`

Rewrite this as: `(2^(a+3) - 2^2*2^a)/ (2^(2a+1) - (2^2)^a)` .

Using the basic rules of exponents : `x^(m+n)=x^m*x^n`   and

`(x^m)^n=x^(mn)` we get:

`(2^a*2^3 - 2^2*2^a)/ (2^(2a)*2 - 2^(2a))`

Factoring out `2^a` from the numerator and `2^(2a)` from the denominator we get:

`=(2^a(2^3-2^2))/(2^(2a)(2-1))`

Now, `2^(2a)=2^a*2^a`

So, cancelling `2^a` from the numerator and the denominator yields:

`(2^3-2^2)/(2^a(2-1))`

`=(8-4)/2^a`

`=4/2^a`

`=2^2/2^a`

`=2^2*2^-a`                     (since, `1/x^m=x^-m` )

`=(2)^(2-a)` .`rArr` answer.

 

 

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