Rewritten Solution

1. Key Concept: The Binomial Theorem and General Term

The problem requires us to find the value of $|\alpha|$ using the concept of the term independent of $x$ in a binomial expansion. The Binomial Theorem states that for any positive integer $n$, the expansion of $(a+b)^n$ is given by: $(a+b)^n = \sum_{r=0}^{n} {^n C_r a^{n-r} b^r}$ The general term, often denoted as $T_{r+1}$, gives any term in the expansion and is expressed as: $T_{r+1} = {^n C_r a^{n-r} b^r}$ A "term independent of $x$" (or constant term) is a term where the variable $x$ has an exponent of zero. This means that after simplifying all $x$ terms, the net power of $x$ must be $0$.

2. Finding the General Term for the Given Expansion

Given the expansion is $\left(x^{\frac{2}{3}}+\frac{\alpha}{x^{3}}\right)^{22}$. Here, we identify:

• $a = x^{\frac{2}{3}}$
• $b = \frac{\alpha}{x^3} = \alpha x^{-3}$
• $n = 22$

Substituting these into the general term formula T_{r+1} = {^n C_r a^{n-r} b^r: $T_{r+1} = {^{22} C_r \left(x^{\frac{2}{3}}\right)^{22-r} \left(\alpha x^{-3}\right)^r}$ Now, we simplify the terms involving $x$ and $\alpha$: $T_{r+1} = {^{22} C_r x^{\frac{2}{3}(22-r)} \alpha^r (x^{-3})^r}$ $T_{r+1} = {^{22} C_r x^{\frac{44-2r}{3}} \alpha^r x^{-3r}}$ To combine the $x$ terms, we add their exponents: $T_{r+1} = {^{22} C_r \alpha^r x^{\frac{44-2r}{3} - 3r}}$ This is the general term of the expansion, expressed in a form that clearly shows the power of $x$.

3. Determining the Value of $r$ for the Term Independent of $x$

For the term to be independent of $x$, the exponent of $x$ must be equal to zero. Therefore, we set the exponent of $x$ from the general term to $0$: $\frac{44-2r}{3} - 3r = 0$ To solve for $r$, we first find a common denominator: $\frac{44-2r - 3(3r)}{3} = 0$ $\frac{44-2r - 9r}{3} = 0$ $44 - 11r = 0$ $11r = 44$ $r = \frac{44}{11}$ $r = 4$ Explanation: We solved for $r$ because the binomial coefficient ${^n C_r}$ requires an integer value for $r$. Since $r=4$ is a non-negative integer and $0 \le r \le n$ (i.e., $0 \le 4 \le 22$), this is a valid value for $r$.

4. Calculating the Constant Term and Finding $\alpha^4$

Now that we have $r=4$, we substitute this value back into the general term, specifically the coefficient part (excluding the $x^0$ term, which is $1$). The term independent of $x$ is $T_{4+1} = T_5$: $T_5 = {^{22} C_4 \alpha^4}$ We are given that this constant term is 7315. So, we set up the equation: ${^{22} C_4 \alpha^4 = 7315}$ Next, we calculate the binomial coefficient ${^{22} C_4}$: ${^{22} C_4 = \frac{22!}{4!(22-4)!} = \frac{22 \times 21 \times 20 \times 19}{4 \times 3 \times 2 \times 1}}$ $= \frac{22 \times 21 \times 20 \times 19}{24}$ $= 11 \times 7 \times 5 \times 19 \quad \text{(after cancelling out common factors)}$ $= 77 \times 95$ $= 7315$ Now, substitute this value back into our equation: $7315 \times \alpha^4 = 7315$ $\alpha^4 = \frac{7315}{7315}$ $\alpha^4 = 1$

5. Finding $|\alpha|$

From the equation $\alpha^4 = 1$, we need to find the real values of $\alpha$. The real solutions for $\alpha$ are $\alpha = 1$ or $\alpha = -1$. The problem asks for the value of $|\alpha|$. For $\alpha = 1$, $|\alpha| = |1| = 1$. For $\alpha = -1$, $|\alpha| = |-1| = 1$. In both cases, $|\alpha| = 1$.

Common Mistake: When solving $\alpha^4 = 1$, it's crucial to remember that $\alpha$ can be positive or negative. If the question had asked for $\alpha$ itself, both $1$ and $-1$ would be valid real answers. However, since it asks for $|\alpha|$, the result is unique.

6. Summary and Key Takeaway

This problem effectively demonstrates the application of the Binomial Theorem's general term formula to find a specific term (in this case, the term independent of $x$). The key steps involved:

1. Correctly identifying $a$, $b$, and $n$ for the general term $T_{r+1} = {^n C_r a^{n-r} b^r}$.
2. Carefully simplifying the powers of $x$ in the general term.
3. Setting the exponent of $x$ to zero to find the value of $r$ for the term independent of $x$.
4. Calculating the binomial coefficient and solving for the unknown variable, $|\alpha|$.

Always remember to check the validity of $r$ (non-negative integer) and pay attention to whether the question asks for the variable itself or its absolute value.