This requires some factoring. We'll factor 2^(x-2) from the left side.Holden123 wrote:If 2^x - 2^(x-2) = 3(2^13), what is the value of x ?
2^x - 2^(x-2) = 3(2^13)
2^(x-2)[2^2 - 1] = 3(2^13)
2^(x-2)[3] = 3(2^13)
So, 2^(x-2)= 2^13
x-2 = 13
[spoiler]x = 15[/spoiler]
Aside: Some students don't see how we can factor 2^(x-2) from the left side. So, let's first examine some analogous factoring cases:
k^5 - k^3 = k^3(k^2 - 1)
m^19 - m^15 = m^15(m^4 - 1)
w^x + x^(x+5) = w^x(1 + w^5)
Notice that, each time, the greatest common factor of both terms is the term with the smallest exponent.
So, in the expression 2^x - 2^(x-2), the term with the smallest exponent is 2^(x-2), so we'll factor out 2^(x-2)
Cheers,
Brent
