If 2^x - 2^(x-2) = 3(2^13), what is the value of x?
a) 9
b) 11
c) 13
d) 15
e) 17
We can plug in the answer choices for x.
Answer choice C: x= 13
2^13 - 2^(13-2) = 3(2^13)
2^13 - 2^11 = 3(2^13)
2^11(2^2 - 1) = 3(2^13)
2^11(3) = 3(2^13)
Plugging in x=13 made the exponent on the left 2^11.
To match 2^13 on the right side of the equation, the exponent needs to be increased by 2.
Thus, x = 13+2 = 15.
The correct answer is
D.
Algebraically:
Try get SIMILAR BASES on each side of the equation.
Given x^y ± x^z, factor out the SMALLER EXPONENT.
2^x - 2^(x-2) = 3(2^13)
2^(x-2)(2^2 - 1) = 3(2^13)
2^(x-2)(3) = 3(2^13)
2^(x-2) = 2^13
x-2 = 13
x = 15.
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