Simplify (4^(x+2)-2^(2x+1))/(8^x (4^(1-x)) and express it in the form ab^(x-2), where a and b are integers?

1 Answer
Jan 2, 2016

14(2^(x-2))

Explanation:

First, write everything in terms of a power of 2.

((2^2)^(x+2)-2^(2x+1))/((2^3)^x((2^2)^(1-x))

Simplify using the rule that (x^a)^b=x^(ab).

(2^(2x+4)-2^(2x+1))/(2^(3x)(2^(2-2x)))

Simplify the denominator using the rule that x^a(x^b)=x^(a+b).

(2^(2x+4)-2^(2x+1))/(2^(x+2))

Split apart the fraction.

(2^(2x+4))/(2^(x+2))-2^(2x+1)/2^(x+2)

Simplify using the rule that x^a/x^b=x^(a-b).

2^(x+2)-2^(x-1)

Factor out a 2^(x-2) term.

2^(x-2)(2^4-2)

Simplify and write in ab^(x-2) form.

14(2^(x-2))