Question

Exponents!

3^(x+2) - 3^(x) = 6^3 (3^7)

Solve for x.

Would very much appreciate a through explaination! (●´∀｀●)

Answer 1

We know that:

$x^{a+b}=x^a\cdot x^b$

and:

$(x\cdot y)^a=x^a\cdot y^a$

so:

$3^{x+2}-3^{x}=6^3\cdot3^7\\\\3^x\cdot3^2-3^x=(2\cdot3)^3\cdot3^7\\\\3^x\cdot9-3^x=2^3\cdot3^3\cdot3^7\\\\3^x\cdot9-3^x=8\cdot3^3\cdot3^7\\\\ 3^x\cdot9-3^x=8\cdot3^{3+7}\\\\3^x\cdot9-3^x=8\cdot3^{10}\\\\3^x\cdot(9-1)=8\cdot3^{10}\\\\8\cdot3^x=8\cdot3^{10}\quad|:8\\\\3^x=3^{10}\\\\\boxed{x=10}$