Expand (3 imes 5x)6 using Pascal's method. If this is not possible, explain why and solve directly. Assume the problem meant (3+5x)6 and expand it.

$$(15x)^6 = 11,390,625x^6$$

The final answer is $\boxed{\text{11,390,625}x^6}$.

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Assuming the problem intended to be $(3+5x)^6$ (a binomial expansion):

If the problem intended to be $(3+5x)^6$, then Pascal's method would be used to find the coefficients for the binomial expansion.

The 6th row of Pascal's Triangle (starting with row 0) provides the coefficients:

$$1, 6, 15, 20, 15, 6, 1$$

The general form for $(a+b)^n$ is $\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$.

• Term 1 ($k=0$): $\binom{6}{0} (3)^6 (5x)^0 = 1 \cdot (729) \cdot (1) = 729$
• Term 2 ($k=1$): $\binom{6}{1} (3)^5 (5x)^1 = 6 \cdot (243) \cdot (5x) = 6 \cdot 1215x = 7290x$
• Term 3 ($k=2$): $\binom{6}{2} (3)^4 (5x)^2 = 15 \cdot (81) \cdot (25x^2) = 15 \cdot 2025x^2 = 30375x^2$