Key Concept: Binomial Theorem and the General Term

The problem asks for coefficients in a binomial expansion. For any binomial expression of the form $(a+b)^n$, the general term, also known as the $(r+1)^{th}$ term ($T_{r+1}$), is given by: $T_{r+1} = {^n C_r a^{n-r} b^r}$ where ${^n C_r = \frac{n!}{r!(n-r)!}}$ is the binomial coefficient. This formula allows us to find any specific term in the expansion without writing out the entire series.

In this specific problem, we are expanding ${\left( {{x^{{1 \over 3}}} + {1 \over {2{x^{{1 \over 3}}}}}} \right)^{18}}$. Comparing this to $(a+b)^n$, we identify:

• $a = x^{1/3}$
• $b = \frac{1}{2x^{1/3}}$
• $n = 18$

Step 1: Derive the General Term ($T_{r+1}$)

Substitute $a$, $b$, and $n$ into the general term formula: $T_{r+1} = {^{18}C_r \left(x^{1/3}\right)^{18-r} \left(\frac{1}{2x^{1/3}}\right)^r}$

Explanation: This is the foundational step. We directly apply the binomial theorem's general term formula, which is designed to give us any term's structure based on its position 'r'.

Step 2: Simplify the General Term to Isolate the Power of $x$

To find the coefficients of specific powers of $x$, we need to simplify $T_{r+1}$ so that all $x$ terms are combined into a single power: $T_{r+1} = {^{18}C_r \cdot x^{\frac{1}{3}(18-r)} \cdot \frac{1}{2^r \cdot x^{\frac{1}{3}r}}}$ $T_{r+1} = {^{18}C_r \cdot \frac{1}{2^r} \cdot x^{6 - \frac{r}{3}} \cdot x^{-\frac{r}{3}}}$ $T_{r+1} = {^{18}C_r \left(\frac{1}{2}\right)^r x^{6 - \frac{r}{3} - \frac{r}{3}}}$ $T_{r+1} = {^{18}C_r \left(\frac{1}{2}\right)^r x^{6 - \frac{2r}{3}}}$ We can rewrite the exponent of $x$ with a common denominator: $T_{r+1} = {^{18}C_r \left(\frac{1}{2}\right)^r x^{\frac{18 - 2r}{3}}}$

Explanation: This step is crucial for identifying which term contains the desired power of $x$. We use exponent rules ($(x^p)^q = x^{pq}$ and $x^p \cdot x^q = x^{p+q}$ or $x^p / x^q = x^{p-q}$) to consolidate all $x$ terms into $x^{\text{single power}}$. The coefficient part ($^{18}C_r (1/2)^r$) is separated as it's what we'll evaluate later.

Tip: Pay close attention to negative exponents and fractions when simplifying. A common error is miscalculating the combined power of $x$.

Step 3: Find the Coefficient of $x^{-2}$ (denoted as $m$)

To find the term containing $x^{-2}$, we set the exponent of $x$ from our general term equal to -2: $\frac{18 - 2r}{3} = -2$ Now, we solve for $r$: $18 - 2r = -6$ $2r = 18 + 6$ $2r = 24$ $r = 12$ When $r=12$, the coefficient of $x^{-2}$ (which is $m$) is the non-$x$ part of $T_{r+1}$: $m = {^{18}C_{12} \left(\frac{1}{2}\right)^{12}}$

Explanation: By equating the calculated power of $x$ to the target power ($-2$), we determine the specific index $r$ that yields that term. This value of $r$ is then substituted into the coefficient part of the general term to find the actual coefficient.

Step 4: Find the Coefficient of $x^{-4}$ (denoted as $n$)

Similarly, to find the term containing $x^{-4}$, we set the exponent of $x$ from our general term equal to -4: $\frac{18 - 2r}{3} = -4$ Now, we solve for $r$: $18 - 2r = -12$ $2r = 18 + 12$ $2r = 30$ $r = 15$ When $r=15$, the coefficient of $x^{-4}$ (which is $n$) is the non-$x$ part of $T_{r+1}$: $n = {^{18}C_{15} \left(\frac{1}{2}\right)^{15}}$

Explanation: This step mirrors Step 3, applying the same method to find the index $r$ and subsequently the coefficient for the second required power of $x$.

Step 5: Calculate the Ratio $m/n$

Now we need to calculate the ratio $m/n$: $\frac{m}{n} = \frac{{^{18}C_{12} \left(\frac{1}{2}\right)^{12}}}{{^{18}C_{15} \left(\frac{1}{2}\right)^{15}}}$

To simplify this expression efficiently, we'll use two key properties:

1. Exponent Rule: $\frac{a^p}{a^q} = a^{p-q}$
2. Binomial Coefficient Property: ${^n C_r = ^n C_{n-r}}$ (This helps reduce the numbers in the factorials, simplifying calculations.)

First, simplify the powers of $1/2$: $\frac{\left(\frac{1}{2}\right)^{12}}{\left(\frac{1}{2}\right)^{15}} = \left(\frac{1}{2}\right)^{12-15} = \left(\frac{1}{2}\right)^{-3} = 2^3 = 8$

Next, apply the binomial coefficient property: ${^{18}C_{12} = ^{18}C_{18-12} = ^{18}C_6}$ ${^{18}C_{15} = ^{18}C_{18-15} = ^{18}C_3}$

Substitute these simplified terms back into the ratio: $\frac{m}{n} = \frac{{^{18}C_6}}{{^{18}C_3}} \cdot 8$

Now, expand the binomial coefficients: $\frac{m}{n} = \frac{\frac{18!}{6!(18-6)!}}{\frac{18!}{3!(18-3)!}} \cdot 8 = \frac{\frac{18!}{6!12!}}{\frac{18!}{3!15!}} \cdot 8$ $\frac{m}{n} = \frac{18!}{6!12!} \cdot \frac{3!15!}{18!} \cdot 8$ The $18!$ terms cancel out: $\frac{m}{n} = \frac{3!15!}{6!12!} \cdot 8$

Expand the larger factorials to cancel terms: $15! = 15 \times 14 \times 13 \times 12!$ $6! = 6 \times 5 \times 4 \times 3!$

$\frac{m}{n} = \frac{3! \times (15 \times 14 \times 13 \times 12!)}{(6 \times 5 \times 4 \times 3!) \times 12!} \cdot 8$ Cancel $3!$ and $12!$: $\frac{m}{n} = \frac{15 \times 14 \times 13}{6 \times 5 \times 4} \cdot 8$ Perform the multiplication and division: $\frac{m}{n} = \frac{(3 \times 5) \times (2 \times 7) \times 13}{(2 \times 3) \times 5 \times (2 \times 2)} \cdot 8$ $\frac{m}{n} = \frac{15 \times 14 \times 13}{120} \cdot 8$ $\frac{m}{n} = \frac{2730}{120} \cdot 8$ $\frac{m}{n} = \frac{273}{12} \cdot 8$ $\frac{m}{n} = \frac{91}{4} \cdot 8$ $\frac{m}{n} = 91 \times 2$ $\frac{m}{n} = 182$

Explanation: This step involves the careful simplification of the ratio. Using the property ${^n C_r = ^n C_{n-r}}$ significantly reduces the complexity of factorial calculations. This makes it easier to cancel common terms and arrive at the final numerical answer.

Common Mistakes to Avoid:

• Incorrectly applying exponent rules, especially with negative and fractional powers.
• Arithmetic errors during factorial expansion or simplification of the ratio.
• Forgetting to use the ${^n C_r = ^n C_{n-r}}$ property, which often simplifies calculations.

Summary and Key Takeaway

To find ratios of coefficients in binomial expansions:

1. Formulate the general term ($T_{r+1}$).
2. Simplify the general term to clearly identify the exponent of $x$.
3. Equate the $x$-exponent to the desired power to solve for $r$.
4. Substitute $r$ back into the non-$x$ part of $T_{r+1}$ to get the specific coefficient.
5. Form the ratio and simplify using exponent rules and binomial coefficient properties like ${^n C_r = ^n C_{n-r}}$ for efficient calculation. The final ratio of the coefficients $m$ and $n$ is 182.