Key Concept: The Binomial Theorem

The binomial theorem provides a formula for expanding expressions of the form $(a+b)^n$. The general term, often denoted as $T_{r+1}$, in the expansion of $(a+b)^n$ is given by: $T_{r+1} = {^n C_r} a^{n-r} b^r$ where $n$ is the power to which the binomial is raised, $r$ is the index of the term (starting from $r=0$), and ${^n C_r} = \frac{n!}{r!(n-r)!}$ is the binomial coefficient.

In this problem, we are given the expression $\left(x^{2 / 3}+\frac{1}{2} x^{-2 / 5}\right)^9$. Comparing this with $(a+b)^n$:

• $a = x^{2/3}$
• $b = \frac{1}{2} x^{-2/5}$
• $n = 9$

Our goal is to find the coefficients of specific powers of $x$ and then sum them.

Step 1: Derive the General Term of the Expansion

First, we substitute the values of $a$, $b$, and $n$ into the general term formula to find the $T_{r+1}$ for our given expansion: $T_{r+1} = {^9 C_r} \left(x^{2/3}\right)^{9-r} \left(\frac{1}{2} x^{-2/5}\right)^r$ To simplify this expression, we apply the rules of exponents, specifically $(x^m)^p = x^{mp}$ and $(xy)^p = x^p y^p$: $T_{r+1} = {^9 C_r} \left(x^{\frac{2}{3}(9-r)}\right) \left(\left(\frac{1}{2}\right)^r \left(x^{-2/5}\right)^r\right)$ $T_{r+1} = {^9 C_r} \frac{1}{2^r} x^{\frac{2(9-r)}{3}} x^{-\frac{2r}{5}}$ Now, we combine the terms with $x$ by adding their exponents, using the rule $x^m \cdot x^p = x^{m+p}$: $T_{r+1} = {^9 C_r} \frac{1}{2^r} x^{\frac{18-2r}{3} - \frac{2r}{5}}$ To add the fractional exponents, we find a common denominator, which is 15: $T_{r+1} = {^9 C_r} \frac{1}{2^r} x^{\frac{5(18-2r) - 3(2r)}{15}}$ $T_{r+1} = {^9 C_r} \frac{1}{2^r} x^{\frac{90-10r - 6r}{15}}$ $T_{r+1} = {^9 C_r} \frac{1}{2^r} x^{\frac{90-16r}{15}}$ This is our general term, where the power of $x$ is $P(r) = \frac{90-16r}{15}$. The coefficient of this term is ${^9 C_r} \frac{1}{2^r}$.

Step 2: Find the Coefficient of $x^{2/3}$

We need to find the value of $r$ for which the power of $x$ in the general term is $2/3$. Set the exponent equal to $2/3$: $\frac{90-16r}{15} = \frac{2}{3}$ To solve for $r$, we can cross-multiply or multiply both sides by the least common multiple of the denominators (which is 15): $15 \cdot \frac{90-16r}{15} = 15 \cdot \frac{2}{3}$ $90-16r = 5 \cdot 2$ $90-16r = 10$ Now, isolate the term with $r$: $16r = 90 - 10$ $16r = 80$ Divide by 16 to find $r$: $r = \frac{80}{16}$ $r = 5$ Now we substitute $r=5$ into the coefficient part of the general term: Coefficient of $x^{2/3} = {^9 C_5} \frac{1}{2^5}$ Calculate ${^9 C_5}$: ${^9 C_5} = \frac{9!}{5!(9-5)!} = \frac{9!}{5!4!} = \frac{9 \times 8 \times 7 \times 6 \times 5!}{5! \times 4 \times 3 \times 2 \times 1} = \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 9 \times 2 \times 7 = 126$ Calculate $2^5 = 32$. So, the coefficient of $x^{2/3}$ is: $126 \times \frac{1}{32} = \frac{126}{32} = \frac{63}{16}$

Step 3: Find the Coefficient of $x^{-2/5}$

Next, we find the value of $r$ for which the power of $x$ in the general term is $-2/5$. Set the exponent equal to $-2/5$: $\frac{90-16r}{15} = -\frac{2}{5}$ Multiply both sides by 15: $15 \cdot \frac{90-16r}{15} = 15 \cdot \left(-\frac{2}{5}\right)$ $90-16r = 3 \cdot (-2)$ $90-16r = -6$ Isolate the term with $r$: $16r = 90 + 6$ $16r = 96$ Divide by 16 to find $r$: $r = \frac{96}{16}$ $r = 6$ Now we substitute $r=6$ into the coefficient part of the general term: Coefficient of $x^{-2/5} = {^9 C_6} \frac{1}{2^6}$ Calculate ${^9 C_6}$: ${^9 C_6} = \frac{9!}{6!(9-6)!} = \frac{9!}{6!3!} = \frac{9 \times 8 \times 7 \times 6!}{6! \times 3 \times 2 \times 1} = \frac{9 \times 8 \times 7}{3 \times 2 \times 1} = 3 \times 4 \times 7 = 84$ Calculate $2^6 = 64$. So, the coefficient of $x^{-2/5}$ is: $84 \times \frac{1}{64} = \frac{84}{64} = \frac{21}{16}$

Step 4: Sum the Coefficients

Finally, we sum the two coefficients we found: Sum $= (\text{Coefficient of } x^{2/3}) + (\text{Coefficient of } x^{-2/5})$ Sum $= \frac{63}{16} + \frac{21}{16}$ Since the denominators are the same, we can directly add the numerators: Sum $= \frac{63+21}{16} = \frac{84}{16}$ Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 4: Sum $= \frac{84 \div 4}{16 \div 4} = \frac{21}{4}$

Tips and Common Mistakes to Avoid:

• Fractional Exponents: Be very careful when handling and combining fractional exponents. A common mistake is arithmetic errors when finding common denominators or multiplying fractions.
• Sign Errors: Double-check the signs, especially when moving terms across the equality sign.
• Binomial Coefficient Calculation: Ensure you correctly calculate ${^n C_r}$. Remember that ${^n C_r} = {^n C_{n-r}}$, which can sometimes simplify calculations (e.g., ${^9 C_6} = {^9 C_3}$).
• Coefficient vs. Term: Remember that the coefficient is the numerical part multiplying $x$, not the entire term $T_{r+1}$.

Summary

We used the general term formula of the binomial expansion to find an expression for the power of $x$ in any term. By equating this expression to the desired powers ($2/3$ and $-2/5$), we found the corresponding values of $r$. Substituting these $r$ values back into the coefficient part of the general term allowed us to calculate the individual coefficients. Finally, summing these coefficients yielded the required result. The sum of the coefficients of $x^{2/3}$ and $x^{-2/5}$ in the given expansion is $\frac{21}{4}$.