If 5x . 25(x+6) = 125(2x+9), then find x.

Concept

When two exponential expressions with different bases can be rewritten using a single common base, and both sides of an equation are then powers of that same base, the two exponents must be equal (since am = an implies m = n, for a > 0, a is not equal to 1). This principle lets an equation involving different powers of 5, 25, and 125 be reduced to a straightforward linear equation once every term is rewritten in terms of base 5.

Applying this to the given equation

1. Rewrite 25 as 52 and 125 as 53, so the equation 5x . 25(x+6) = 125(2x+9) becomes 5x . (52)(x+6) = (53)(2x+9).

2. Apply the power-of-a-power rule (am)n = amn to each side: 5x . 5(2x+12) = 5(6x+27).

3. Apply the product rule am . an = am+n on the left side: 5(x+2x+12) = 5(6x+27), that is, 5(3x+12) = 5(6x+27).

4. Since both sides now share base 5, equate the exponents: 3x + 12 = 6x + 27.

5. Collect the x-terms on one side and the constants on the other: 3x - 6x = 27 - 12, that is, -3x = 15.

6. Divide both sides by -3: x = -5.

Cross-check

Substituting x = -5 back into the original equation confirms the result: 5(-5) . 251 = 5(-5) . 52 = 5(-3) = 1/125, and 125(2(-5)+9) = 125(-1) = 1/125 - both sides match, verifying x = -5.