The expansion of \((a+b)^5\) can be done using the **Binomial Theorem**, which states:

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

For \(n = 5\), the binomial expansion of \((a+b)^5\) is:

\[
(a + b)^5 = \binom{5}{0} a^5 + \binom{5}{1} a^4 b + \binom{5}{2} a^3 b^2 + \binom{5}{3} a^2 b^3 + \binom{5}{4} a b^4 + \binom{5}{5} b^5
\]

Calculating the binomial coefficients \(\binom{5}{k}\):

\[
\binom{5}{0} = 1, \quad \binom{5}{1} = 5, \quad \binom{5}{2} = 10, \quad \binom{5}{3} = 10, \quad \binom{5}{4} = 5, \quad \binom{5}{5} = 1
\]

Substituting these values:

\[
(a + b)^5 = a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5
\]

So, the expanded form of \((a + b)^5\) is:

\[
a^5 + 5a^4b + 10a^3b^2 + 10a^2b^3 + 5ab^4 + b^5
\]

Would you like more details or have any questions?

Here are 5 related questions to expand on this:

1. How do you expand \((a-b)^5\) using the binomial theorem?
2. What is the binomial coefficient \(\binom{6}{2}\)?
3. How do you use Pascal’s triangle to expand binomials?
4. How do you find the middle term in the expansion of \((a + b)^6\)?
5. Can you derive the general formula for the binomial expansion of \((x + y)^n\)?

**Tip:** For higher powers, Pascal's triangle provides a quick way to identify binomial coefficients.

Explore the detailed expansion of (a + b)⁵ using the binomial theorem. Learn how to apply binomial coefficients and expand polynomials step-by-step. Ideal for students in Grades 9-12.

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