Final answer:
The given algebraic expression is simplified by dividing the common factors, subtracting the exponents of like bases, and finally ensuring that units cancel out correctly to arrive at the simplified expression 7/10 * u² * (u+3)² / (3u-7).
Step-by-step explanation:
To simplify the expression (28u³(u+3)⁵)/(40u(u+3)³(3u-7)), we first identify and eliminate any common factors in the numerator and denominator. Then, we apply the rule for division of exponentials, which involves subtracting the exponents of like bases. Let's break down the simplification step by step:
- Simplify the coefficients: Divide 28 by 40 to get 7/10.
- Simplify u terms: u³ / u = u², since we subtract the exponent in the denominator from the exponent in the numerator (3 - 1 = 2).
- Simplify (u+3) terms: (u+3)⁵ / (u+3)³ = (u+3)², for the same reason.
- Notice that (3u-7) does not share a common factor with other terms, so it remains as is in the denominator.
After simplifying, we have 7/10 * u² * (u+3)² / (3u-7).
Lastly, we check that this is a reasonable simplification. The units of the original expression have canceled out correctly, leaving us with a simplified version of our original algebraic fraction.