In the equation 2(x+5) = 2(x+3) + 12, the value of x is

Concept: For an exponential equation whose terms share a common base, rewrite each term using the law a(m+n) = am × an, collect the like powers of the base into a single equation, and then equate the resulting powers of the same base to solve for the unknown exponent (if ap = aq then p = q).

Application:

1. Expand each term using the product law: 2(x+5) = 2x × 25 = 32 × 2x, and 2(x+3) = 2x × 23 = 8 × 2x.

2. Substitute these into the given equation: 32 × 2x = 8 × 2x + 12.

3. Collect the powers of 2x on one side: 32 × 2x - 8 × 2x = 12, so 24 × 2x = 12.

4. Isolate 2x: 2x = 12 / 24 = 1/2 = 2-1.

5. Equate the exponents of the same base 2: x = -1.

Cross-check: Substituting x = -1 back into the original terms gives 2(-1+5) = 24 = 16 on the left, and 2(-1+3) + 12 = 22 + 12 = 4 + 12 = 16 on the right - both sides equal 16, confirming x = -1.