Find the 4th term in the expansion of (a+b)^3 In simplest form

The expression given is \((a + b)^3\), and we need to find the 4th term in its expansion.

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 \((a + b)^3\), the expansion will be:

\[
(a + b)^3 = \binom{3}{0}a^3b^0 + \binom{3}{1}a^2b^1 + \binom{3}{2}a^1b^2 + \binom{3}{3}a^0b^3
\]

Simplifying, we get:

\[
(a + b)^3 = 1 \cdot a^3 + 3 \cdot a^2b + 3 \cdot ab^2 + 1 \cdot b^3
\]

So, the terms of the expansion are:

1. \(a^3\)
2. \(3a^2b\)
3. \(3ab^2\)
4. \(b^3\)

Therefore, the 4th term in the expansion is \(\mathbf{b^3}\).

Would you like details on binomial coefficients, or have any questions? 

Here are 5 related questions:
1. What is the 3rd term in the expansion of \((a - b)^4\)?
2. How do you calculate the binomial coefficient \(\binom{n}{k}\)?
3. What is the general term formula in the binomial expansion of \((a + b)^n\)?
4. How can the binomial theorem be applied to solve problems in probability?
5. How would the expansion differ if the expression were \((a + b)^5\)?

**Tip:** In binomial expansions, each term is structured as a product of a binomial coefficient, one variable raised to a power, and the other variable raised to the complementary power.

Learn how to find the 4th term in the expansion of (a+b)^3 with a detailed explanation and step-by-step solution. This problem involves applying the binomial theorem and understanding binomial coefficients. Suitable for high school students and anyone interested in algebraic expansions.

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