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The expression given is \((3x - 1)^8\), and you are asked to find the fifth term when this expression is expanded.

We can use the **Binomial Theorem** to expand \((a + b)^n\). The general term for the expansion of \((a + b)^n\) is:

\[
T_{k+1} = \binom{n}{k} a^{n-k} b^k
\]

In this case, we have:
- \( a = 3x \)
- \( b = -1 \)
- \( n = 8 \)

The fifth term corresponds to \( k = 4 \) (since the first term is for \( k = 0 \)).

### Formula for the fifth term:

\[
T_5 = \binom{8}{4} (3x)^{8-4} (-1)^4
\]

1. **Binomial Coefficient**: \(\binom{8}{4} = \frac{8!}{4!(8-4)!} = 70\)

2. **Power of \(a\)**: \((3x)^4 = 3^4 x^4 = 81x^4\)

3. **Power of \(b\)**: \((-1)^4 = 1\)

### Fifth Term Calculation:

\[
T_5 = 70 \times 81x^4 \times 1 = 5670x^4
\]

Thus, the fifth term of the expansion is:

\[
\boxed{5670x^4}
\]

Would you like a detailed explanation of the binomial theorem or have any further questions?

Here are five related questions for further exploration:
1. How do you find the third term in \((3x - 1)^8\)?
2. Can you expand \((2x + 5)^6\) using the binomial theorem?
3. How is the binomial coefficient \(\binom{n}{k}\) derived?
4. What is the general form of the seventh term in the expansion of \((x - 4)^9\)?
5. How do powers of negative numbers affect terms in a binomial expansion?

**Tip**: Always check the sign of the terms carefully when expanding expressions involving negative numbers!

Find the fifth term of the expansion of (3x - 1)^8.

Find the fifth term of the expansion of (3x - 1)^8 using the binomial theorem. This problem covers key algebraic concepts such as the binomial expansion, calculation of binomial coefficients, and power of terms. Suitable for students in Grades 10-12.

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