Rewrite all terms with the same base.

Question

ScanSolve AI Tutor · Accepted Answer

Here's how to solve the exponential equation:

Rewrite all terms with the same base.
The base 2 is already present. We need to express 8 and 128 as powers of 2.
$8 = 2^3$ $128 = 2^7$
Substitute these into the equation:
$2^{(x-3)} \times (2^3)^{(x+2)} = 2^7$
Apply the exponent rule $(a^m)^n = a^{mn}$ .
$2^{(x-3)} \times 2^{3(x+2)} = 2^7$ $2^{(x-3)} \times 2^{(3x+6)} = 2^7$
Apply the exponent rule $a^m \times a^n = a^{m+n}$ .
$2^{(x-3) + (3x+6)} = 2^7$ $2^{x-3+3x+6} = 2^7$ $2^{4x+3} = 2^7$
Equate the exponents since the bases are the same.
$4x+3 = 7$
Solve the linear equation for $x$ .
$4x = 7-3$ $4x = 4$ $x = \frac{4}{4}$ $x = 1$

Answer