Problem 52 - Beginning and Intermediate Algebra

Question

Find the indicated term of each binomial expansion. \((y+4)^{7} ;\) fifth term

Step-by-Step Solution

Answer

The fifth term of the binomial expansion \((y+4)^7\) is \(\boxed{8960y^3}\).

Step 1: Identify the binomial terms and the term index

From the given expression, we identify: - \(a = y\) - \(b = 4\) - \(n = 7\) - \(k = 4\)

Step 2: Use the binomial theorem formula

Now plug the values of \(a\), \(b\), \(n\), and \(k\) into the binomial theorem formula: \[\binom{n}{k}a^{n-k}b^k = \binom{7}{4}y^{7-4}4^4\]

Step 3: Calculate the binomial coefficient

Calculate the binomial coefficient using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\): \[\binom{7}{4} = \frac{7!}{4!(3)!} = \frac{7\times6\times5}{4\times3\times2} = 35\]

Step 4: Calculate the y exponent

Subtract \(k\) from \(n\): \[7 - 4 = 3\]

Step 5: Calculate the power of 4

Calculate \(4^4\): \[4^4 = 256\]

Step 6: Combine the results

Now, combine the results of Step 3, Step 4, and Step 5: \[35 \cdot y^3 \cdot 256\] So, the fifth term of the binomial expansion \((y+4)^7\) is \(\boxed{8960y^3}\).