We are asked to expand $(4 + 3y)^3$ using Pascal's Triangle.

Step-by-step process:

1. Coefficients from Pascal's Triangle: The third row of Pascal's Triangle (since we are raising to the power of 3) is:
   $1, 3, 3, 1$

2. Expand using binomial terms: The binomial expansion of $(a + b)^n$ is given by:
   $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$ For $(4 + 3y)^3$, we substitute $a = 4$, $b = 3y$, and $n = 3$: $(4 + 3y)^3 = 1(4)^3(3y)^0 + 3(4)^2(3y)^1 + 3(4)^1(3y)^2 + 1(4)^0(3y)^3$

3. Calculate each term:

   • $1(4)^3(3y)^0 = 1 \cdot 64 \cdot 1 = 64$
   • $3(4)^2(3y)^1 = 3 \cdot 16 \cdot 3y = 144y$
   • $3(4)^1(3y)^2 = 3 \cdot 4 \cdot 9y^2 = 108y^2$
   • $1(4)^0(3y)^3 = 1 \cdot 1 \cdot 27y^3 = 27y^3$

4. Combine all terms: $(4 + 3y)^3 = 64 + 144y + 108y^2 + 27y^3$

Thus, the expanded form is: $64 + 144y + 108y^2 + 27y^3$

Would you like more details or have any questions?

Relative Questions:

1. How can you find any row of Pascal's Triangle without constructing the whole triangle?
2. What is the binomial expansion formula in general terms?
3. How does Pascal’s Triangle help in binomial expansions?
4. How would the expansion differ if the power was increased to 4 instead of 3?
5. What happens when the coefficients of $x$ and $y$ terms are negative?

Tip: When using Pascal's Triangle, remember that the exponents of the terms $a$ and $b$ decrease and increase respectively, with each successive term.