Final answer:
The expansion of (3z+3y)³ using Pascal's Triangle and the binomial theorem is 27z³ + 81z²y + 81zy² + 27y³.
Step-by-step explanation:
To expand the expression (3z+3y)³ using Pascal's Triangle, we will use the third row which corresponds to the coefficients for the expansion of a binomial raised to the third power. The third row of Pascal's Triangle is 1, 3, 3, 1.
The binomial theorem tells us that the expansion will have four terms, and using the coefficients from Pascal's Triangle, the expansion is obtained as follows:
The first term: 1 × (3z)³ = 27z³
The second term: 3 × (3z)²(3y) = 81z²y
The third term: 3 × (3z)(3y)² = 81zy²
The final term: 1 × (3y)³ = 27y³
Combining these terms, the expanded expression in simplest form is 27z³ + 81z²y + 81zy² + 27y³.