Expanding (x + 4)^4
Expanding expressions with exponents can be tedious, but there are methods to make the process more manageable. Here's a breakdown of how to expand (x + 4)^4:
Understanding the Binomial Theorem
The most efficient way to expand such expressions is through the Binomial Theorem. This theorem states that for any positive integer n, the expansion of (x + y)^n is given by:
(x + y)^n = ∑(n choose k) * x^(nk) * y^k
Where:
 ∑ represents the summation from k = 0 to n
 (n choose k) is the binomial coefficient, calculated as n! / (k! * (nk)!)
Applying the Binomial Theorem to (x + 4)^4

Identify n: In our case, n = 4.

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

Substitute into the formula:
(x + 4)^4 = (1 * x^4 * 4^0) + (4 * x^3 * 4^1) + (6 * x^2 * 4^2) + (4 * x^1 * 4^3) + (1 * x^0 * 4^4)

Simplify:
(x + 4)^4 = x^4 + 16x^3 + 96x^2 + 256x + 256
Other Methods
While the Binomial Theorem is the most elegant approach, you can also expand the expression by repeated multiplication:
 (x + 4)^2 = (x + 4) * (x + 4) = x^2 + 8x + 16
 (x + 4)^3 = (x^2 + 8x + 16) * (x + 4) = x^3 + 12x^2 + 48x + 64
 (x + 4)^4 = (x^3 + 12x^2 + 48x + 64) * (x + 4) = x^4 + 16x^3 + 96x^2 + 256x + 256
Conclusion
The Binomial Theorem offers a powerful and concise method for expanding expressions of the form (x + y)^n. Regardless of the method chosen, understanding the process of expanding such expressions is crucial for various mathematical applications.