Problem 55 - College Algebra

Question

What is true about the sum of the exponents on \(a\) and \(b\) in any term in the expansion of \((a+b)^{n} ?\)

Step-by-Step Solution

Answer

In the expansion of \((a+b)^{n}\), the sum of the exponents on \(a\) and \(b\) in any term is always \(n\).

Step 1: Understanding the Binomial Theorem

The binomial theorem (or binomial expansion) describes the algebraic expansion of powers of a binomial. In other words, it tells us how to expand expressions of the form \((a+b)^{n}\). The general term in the expansion of \((a+b)^{n}\) as per the binomial theorem is given by \(nCk \cdot a^{n-k} \cdot b^{k}\), where \(nCk\) gives the binomial coefficient, \(a^{n-k}\) is the \(n-k\)th power of \(a\), and \(b^{k}\) is the \(k\)th power of \(b\), with \(k\) taking all integer values from 0 to \(n\).

Step 2: Applying the theorem to our problem

Let's take a look at the exponents part of this general term. For \(a\) we have the exponent \(n - k\) and for \(b\) we have the exponent \(k\). If we add these exponents together, we get \((n - k) + k = n\). This result implies that for any term in the expansion of \((a+b)^{n}\), the sum of the exponents on \(a\) and \(b\) is always equal to \(n\).

Step 3: Exemplifying the theorem

To illustrate this, if we take the third term from the expansion of \((a+b)^{5}\), which is \(10a^{2}b^{3}\), we find that when we add the exponents on \(a\) and \(b\), \(2 + 3\), we indeed get 5, which is our \(n\) value.