Problem 51 - Beginning and Intermediate Algebra

Question

Find the indicated term of each binomial expansion. \((k+5)^{8} ;\) third term

Step-by-Step Solution

Answer

The third term of the binomial expansion \((k+5)^8\) is \(700k^6\).

Step 1: Identify the Variables

In our binomial expansion, we have a=k, b=5, and n=8. Our target term has k=2.

Step 2: Calculate the Binomial Coefficient

Using the formula for the binomial coefficient, we have: \(\binom{n}{k}=\binom{8}{2}=\frac{8!}{2!(8-2)!}=\frac{8!}{2!6!}\) Now, calculate the factorials: \(= \frac{8 \times 7 \times 6!}{2! \times 6!}\) Since 6! cancels out from the numerator and denominator, we are left with: \(= \frac{8 \times 7}{2}\) Finally, calculate the value: \(= 4 \times 7 = 28\) The binomial coefficient for the third term is 28.

Step 3: Calculate the Powers of a and b

Recall that for the third term, we are using k=2. We need to find the powers of a (k) and b (5): \(a^{n-k} = k^{8-2} = k^6\) \(b^k = 5^2 = 25\)

Step 4: Find the Third Term

Now that we have the binomial coefficient and the powers of a and b, we can find the third term by multiplying these values together: \(T_3 = \binom{n}{k} \times a^{n-k} \times b^k = 28 \times k^6 \times 25\) Simplify the expression: \(T_3 = 700k^6\) The third term of the binomial expansion (k+5)^8 is \(700k^6\).