Expanding (m+n)^4: A Guide Using the Binomial Theorem
The expression (m+n)^4 represents the fourth power of the binomial (m+n). Expanding this expression involves multiplying the binomial by itself four times. While you can do this directly, it becomes tedious for higher powers. The Binomial Theorem offers a much more efficient way to expand such expressions.
Understanding the Binomial Theorem
The Binomial Theorem provides a formula for expanding expressions of the form (x+y)^n, where n is a nonnegative integer. The theorem states:
(x + y)^n = ∑_(k=0)^n (n choose k) * x^(nk) * y^k
Where:
 (n choose k) represents the binomial coefficient, calculated as n! / (k! * (nk)!). This represents the number of ways to choose k items from a set of n items.
 ∑_(k=0)^n represents the sum of all terms from k=0 to k=n.
Expanding (m+n)^4 using the Binomial Theorem

Identify n and the terms: In our case, n = 4, x = m, and y = n.

Calculate the binomial coefficients: We need to calculate the binomial coefficients for k = 0, 1, 2, 3, and 4.
 (4 choose 0) = 4! / (0! * 4!) = 1
 (4 choose 1) = 4! / (1! * 3!) = 4
 (4 choose 2) = 4! / (2! * 2!) = 6
 (4 choose 3) = 4! / (3! * 1!) = 4
 (4 choose 4) = 4! / (4! * 0!) = 1

Apply the Binomial Theorem: Substitute the values into the formula:
(m + n)^4 = (4 choose 0) * m^4 * n^0 + (4 choose 1) * m^3 * n^1 + (4 choose 2) * m^2 * n^2 + (4 choose 3) * m^1 * n^3 + (4 choose 4) * m^0 * n^4

Simplify the expression:
(m + n)^4 = m^4 + 4m^3n + 6m^2n^2 + 4mn^3 + n^4
Key Observations
 The expansion of (m+n)^4 results in a polynomial with 5 terms.
 The exponents of m decrease from 4 to 0, while the exponents of n increase from 0 to 4.
 The coefficients follow a symmetrical pattern: 1, 4, 6, 4, 1, which are the elements of Pascal's Triangle for the fourth row.
Conclusion
The Binomial Theorem provides a powerful tool for expanding binomial expressions raised to any power. Applying the theorem to (m+n)^4, we arrive at the expanded form m^4 + 4m^3n + 6m^2n^2 + 4mn^3 + n^4, demonstrating the efficiency and elegance of the theorem.