What is the constant term in the expansion of $\left(6x+\dfrac{1}{3x}\right)^8$?

(The constant term is the term without a variable. For example, the constant term of $3a+2+\dfrac{1}{a}$ is $2.$)

To find the constant term in the expansion of $\left(6x + \frac{1}{3x}\right)^8$, we can use the Binomial Theorem, which states that:

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

In our case, let $a = 6x$ and $b = \frac{1}{3x}$, and $n = 8$. The general term in the expansion can be written as:

\[
T_k = \binom{8}{k} (6x)^{8-k} \left(\frac{1}{3x}\right)^k
\]

This simplifies to:

\[
T_k = \binom{8}{k} (6^{8-k}) (x^{8-k}) \left(\frac{1}{3^k x^k}\right) = \binom{8}{k} \frac{6^{8-k}}{3^k} x^{8-k-k} = \binom{8}{k} \frac{6^{8-k}}{3^k} x^{8-2k}
\]

We want to find the constant term, which occurs when the exponent of $x$ is zero:

\[
8 - 2k = 0
\]

Solving for $k$:

\[
8 = 2k \implies k = 4
\]

Now, we substitute $k = 4$ back into the expression for $T_k$:

\[
T_4 = \binom{8}{4} \frac{6^{8-4}}{3^4} x^{8-2 \cdot 4}
\]

Calculating this step by step:

1. Calculate $\binom{8}{4}$:

\[
\binom{8}{4} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70
\]

2. Calculate $6^{8-4} = 6^4$:

\[
6^4 = 1296
\]

3. Calculate $3^4$:

\[
3^4 = 81
\]

Now, substituting these values into $T_4$:

\[
T_4 = 70 \cdot \frac{1296}{81}
\]

Calculating $\frac{1296}{81}$:

\[
\frac{1296}{81} = 16
\]

Thus,

\[
T_4 = 70 \cdot 16 = 1120
\]

The constant term in the expansion of $\left(6x + \frac{1}{3x}\right)^8$ is:

\[
\boxed{1120}
\] 

If you have any questions or would like further details, let me know! Here are some related questions you might consider:

1. How would you find the constant term in a different binomial expansion, such as $(2x + 5)^6$?
2. What is the role of the Binomial Theorem in polynomial expansions?
3. How do the coefficients in the expansion relate to the original terms?
4. Can you explain how to derive the general term in a binomial expansion?
5. What would happen to the constant term if the power of the binomial were increased to 10? 

**Tip:** Always check the exponent conditions to find the specific term you need in polynomial expansions!

What is the constant term in the expansion of (6x + 1/3x)^8?

Discover how to find the constant term in the binomial expansion of (6x + 1/3x)^8 using the Binomial Theorem. This detailed guide walks you through the process, including necessary calculations and explanations. Ideal for high school students.

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