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