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}`