If 5(a + b) = 5 × 25 × 125, what is (a + b)2?

Concept: Law of exponents (product rule) — for the same base, multiplying powers adds their exponents: am × an = am + n. Also, if two powers with the same base are equal, their exponents must be equal: am = an ⇒ m = n (for a > 0, a ≠ 1).

1. Rewrite 25 and 125 as powers of base 5: 25 = 52, and 125 = 53.

2. Substitute into the equation: 5(a + b) = 51 × 52 × 53.

3. Apply the product rule to combine the right-hand side into a single power of 5: 5(a + b) = 51 + 2 + 3 = 56.

4. Since the base (5) is the same on both sides, equate the exponents: a + b = 6.

Independent check: substituting a + b = 6 back gives 56 = 15625, and 5 × 25 × 125 = 15625 too — confirming a + b = 6.

Therefore, (a + b)2 = 62 = 36.