Question

How do you solve the exponential equation `2^(x+2)=16^x`?

Answer 1

`x=2/3`

Explanation:

Note that `16=2^4`

`rArr2^(x+2)=(2^4)^x`

`rArrcolor(red)(2)^(x+2)=color(red)(2)^(4x)`

Since the bases are equal, that is `color(red)(2)` then the exponents are equal.

`"solve "4x=x+2`

`rArr3x=2`

`rArrx=2/3" is the solution"`

Answer 2

`x=2/3`

Explanation:

`2^(x+2)=16^x`

`2^(x+2)=(2*2*2*2)^x`

`2^(x+2)=(2^4)^x`

`2^(x+2)=2^(4x)`

because the base on the LHS =2 and base on RHS=2,
so the exponents are equal

`x+2=4x`

`x-4x=-2`

`-3x=-2`

multiply both sides by `-1`

`3x=2`

`x=2/3`

substitute`x=2/3`

`2^((2/3)+2)=16^(2/3)`

`2^(8/3)=(2*2*2*2)^(2/3)`

`2^(8/3)=(2^4)^(2/3)`

`2^(8/3)=2^(8/3)`