Question

Solve for x: `2^(3x + 4) = 2^(3x + 2) + 96`.

Answer 1

We need to solve:

`2^(3x + 4) = 2^(3x + 2) + 96`

Step 1: Factor powers of 2

Notice `2^(3x + 4) = 2^(3x + 2) * 2^2`:

`2^(3x + 4) = 4 * 2^(3x + 2)`

So the equation becomes:

`4 * 2^(3x + 2) = 2^(3x + 2) + 96`

Step 2: Rearrange

`4 * 2^(3x + 2) - 2^(3x + 2) = 96`

`3 * 2^(3x + 2) = 96`

Step 3: Simplify

`2^(3x + 2) = 96/3 = 32`

Step 4: Write as power of 2

`32 = 2^5`

So, `2^(3x + 2) = 2^5`

⇒ `3x + 2 = 5`

Step 5: Solve for x

`3x = 3`

⇒ `x = 1`