To expand $(2a - 3b)^4$ using Pascal's triangle, we follow these steps:

Step 1: Write down the coefficients from Pascal's Triangle for $n = 4$

For $n = 4$, the row in Pascal's triangle is: $1, 4, 6, 4, 1$

Step 2: Apply the Binomial Theorem

The binomial theorem states: $(x + y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$ Here, $x = 2a$, $y = -3b$, and $n = 4$.

Step 3: Expand the terms

Substitute the coefficients and calculate each term:

1. First term ($k = 0$): $\binom{4}{0} (2a)^4 (-3b)^0 = 1 \cdot (16a^4) \cdot 1 = 16a^4$
2. Second term ($k = 1$): $\binom{4}{1} (2a)^3 (-3b)^1 = 4 \cdot (8a^3) \cdot (-3b) = -96a^3b$
3. Third term ($k = 2$): $\binom{4}{2} (2a)^2 (-3b)^2 = 6 \cdot (4a^2) \cdot (9b^2) = 216a^2b^2$
4. Fourth term ($k = 3$): $\binom{4}{3} (2a)^1 (-3b)^3 = 4 \cdot (2a) \cdot (-27b^3) = -216ab^3$
5. Fifth term ($k = 4$): $\binom{4}{4} (2a)^0 (-3b)^4 = 1 \cdot 1 \cdot (81b^4) = 81b^4$

Step 4: Combine all terms

$(2a - 3b)^4 = 16a^4 - 96a^3b + 216a^2b^2 - 216ab^3 + 81b^4$

Would you like a step-by-step explanation of Pascal's triangle or the binomial theorem?

5 Related Questions

1. How do you generate Pascal's triangle for higher powers?
2. What is the significance of the binomial theorem in mathematics?
3. How does the sign of $b$ affect the expansion of a binomial?
4. Can the expansion of $(2a - 3b)^4$ be verified by multiplying $(2a - 3b)$ four times?
5. How would this expansion differ for $(2a + 3b)^4$?

Tip

Always double-check the signs and coefficients when expanding binomials to avoid common mistakes!