To expand $(a + b)^4$, we use the Binomial Theorem. The general formula for expanding $(a + b)^n$ is:

$(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$

where $\binom{n}{k}$ are the binomial coefficients.

For $(a + b)^4$, this becomes:

$(a + b)^4 = \sum_{k=0}^{4} \binom{4}{k} a^{4-k} b^k$

Step-by-Step Expansion

1. Compute the binomial coefficients $\binom{4}{k}$:

   • $\binom{4}{0} = 1$
   • $\binom{4}{1} = 4$
   • $\binom{4}{2} = 6$
   • $\binom{4}{3} = 4$
   • $\binom{4}{4} = 1$

2. Apply the binomial theorem:

   $(a + b)^4 = \binom{4}{0} a^4 b^0 + \binom{4}{1} a^3 b^1 + \binom{4}{2} a^2 b^2 + \binom{4}{3} a^1 b^3 + \binom{4}{4} a^0 b^4$

3. Simplify the terms:

   $(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4$

Final Expansion

$(a + b)^4 = a^4 + 4a^3b + 6a^2b^2 + 4ab^3 + b^4$

Would you like a detailed explanation of each step?

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Related Questions

1. How do you expand $(a + b)^5$ using the binomial theorem?
2. What is the general expression for $(a + b)^n$?
3. How do you find the binomial coefficient $\binom{n}{k}$?
4. What are Pascal’s Triangle and its relation to binomial expansion?
5. How can you expand $(2x + 3)^4$ using the binomial theorem?

Tip

The binomial theorem is useful for expanding powers of binomials and can also be applied to approximate calculations using series expansions.