Answer :
To find the fifth term in the binomial expansion of [tex]\((x+5)^8\)[/tex], we use the binomial expansion formula:
[tex]\[
T(k) = \binom{n}{k} \cdot a^{(n-k)} \cdot b^k
\][/tex]
Here, [tex]\(\binom{n}{k}\)[/tex] represents the binomial coefficient, where [tex]\(n\)[/tex] is the power of the binomial, and [tex]\(k\)[/tex] is the term number minus one because binomial expansion starts counting terms from zero.
Let's go through the steps:
1. Identify the values:
- [tex]\(n = 8\)[/tex] (the power of the binomial)
- The 5th term means [tex]\(k = 4\)[/tex] (because indexing starts at zero)
- In the binomial [tex]\((x+5)\)[/tex], [tex]\(a = x\)[/tex] and [tex]\(b = 5\)[/tex].
2. Calculate the binomial coefficient:
- Use [tex]\(\binom{n}{k} = \binom{8}{4}\)[/tex]. The binomial coefficient can be calculated as:
[tex]\[
\binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70
\][/tex]
3. Calculate the term using the formula:
- Substitute the values into the term formula:
[tex]\[
T(5) = \binom{8}{4} \cdot x^{(8-4)} \cdot 5^4
\][/tex]
- Simplify [tex]\(x^{(8-4)}\)[/tex] to [tex]\(x^4\)[/tex].
- Now evaluate [tex]\(5^4\)[/tex]:
[tex]\[
5^4 = 5 \times 5 \times 5 \times 5 = 625
\][/tex]
- Multiply the components:
[tex]\[
70 \cdot x^4 \cdot 625
\][/tex]
[tex]\[
= 70 \cdot 625 \cdot x^4 = 43750 \cdot x^4
\][/tex]
Therefore, the fifth term in the binomial expansion of [tex]\((x+5)^8\)[/tex] is [tex]\(43,750 x^4\)[/tex].
[tex]\[
T(k) = \binom{n}{k} \cdot a^{(n-k)} \cdot b^k
\][/tex]
Here, [tex]\(\binom{n}{k}\)[/tex] represents the binomial coefficient, where [tex]\(n\)[/tex] is the power of the binomial, and [tex]\(k\)[/tex] is the term number minus one because binomial expansion starts counting terms from zero.
Let's go through the steps:
1. Identify the values:
- [tex]\(n = 8\)[/tex] (the power of the binomial)
- The 5th term means [tex]\(k = 4\)[/tex] (because indexing starts at zero)
- In the binomial [tex]\((x+5)\)[/tex], [tex]\(a = x\)[/tex] and [tex]\(b = 5\)[/tex].
2. Calculate the binomial coefficient:
- Use [tex]\(\binom{n}{k} = \binom{8}{4}\)[/tex]. The binomial coefficient can be calculated as:
[tex]\[
\binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70
\][/tex]
3. Calculate the term using the formula:
- Substitute the values into the term formula:
[tex]\[
T(5) = \binom{8}{4} \cdot x^{(8-4)} \cdot 5^4
\][/tex]
- Simplify [tex]\(x^{(8-4)}\)[/tex] to [tex]\(x^4\)[/tex].
- Now evaluate [tex]\(5^4\)[/tex]:
[tex]\[
5^4 = 5 \times 5 \times 5 \times 5 = 625
\][/tex]
- Multiply the components:
[tex]\[
70 \cdot x^4 \cdot 625
\][/tex]
[tex]\[
= 70 \cdot 625 \cdot x^4 = 43750 \cdot x^4
\][/tex]
Therefore, the fifth term in the binomial expansion of [tex]\((x+5)^8\)[/tex] is [tex]\(43,750 x^4\)[/tex].
