How can the binomial formula be applied to expand x raised to the 13th power?

The binomial theorem provides a powerful way to expand expressions of the form (a + b)ⁿ. Although we are specifically looking to expand x¹³, we can think of this expression in the context of the binomial theorem by letting a = x and b = 0 since we want to express x¹³ as a binomial.

To apply the binomial theorem:

(a + b)n = ∑k=0n C(n, k) * an-k * bk

Where C(n, k) is the binomial coefficient calculated as:

C(n, k) = n! / (k!(n - k)!)

In this case, setting a = x, b = 0, and n = 13, we get:

(x + 0)13 = ∑k=013 C(13, k) * x13-k * 0k

Since any term multiplied by zero will equal zero, the only term that contributes to this expansion occurs when k = 0. Therefore:

C(13, 0) * x13 * 00 = 1 * x13 * 1 = x13

Thus, the result of expanding x¹³ using the binomial theorem is simply:

x13

In summary, because we only have x raised to the 13th power and zero as the other term, expanding it using the binomial formula does not yield a complex result—it straightforwardly results in x¹³. This demonstrates a case where the binomial expansion simplifies rather than complicates the expression.