Final answer:
The simplified form of the given algebraic expression is (2w(w+7)²) / (5(3w-8)) after cancelling out the common w and (w+7) terms, and simplifying the numeric coefficients.
Step-by-step explanation:
To simplify the algebraic expression (18w⁴ (w+7)⁵)/(45w³ (w+7)³(3w-8)), we can cancel out common factors in the numerator and denominator.
Firstly, we notice that both the numerator and denominator have w terms and (w+7) terms, so we divide both parts by their common factors:
- The w³ term in the denominator cancels out three w's in the numerator, leaving us with just w in the numerator.
- The (w+7)³ term in the denominator cancels out three (w+7)'s in the numerator, leaving us with (w+7)² in the numerator.
Secondly, we notice that 18 and 45 are both divisible by 9, so when we simplify 18/45, we get 2/5.
Putting it all together, the simplified expression is:
(2w(w+7)²) / (5(3w-8))