Question

If \( |x| < \frac{2}{3} \), then the fourth term in the expansion of \( (3x - 2)^{2/3} \) is:

• A. \( \frac{\sqrt[3]{4}}{6} x^3 \)
• B. \( -\frac{\sqrt[3]{4}}{6} x^3 \)
• C. \( \frac{\sqrt[3]{4}}{8} x^3 \)
• D. \( -\frac{\sqrt[3]{4}}{8} x^3 \)

Hint:

When dealing with binomial expansions involving negative or fractional exponents, focus on calculating each term's components carefully, especially the powers and coefficients.

Answer 1

The Correct Option is B

Step 1: Apply the binomial theorem for fractional exponents to the expression \( (3x - 2)^{2/3} \).
- The general term in the expansion is given by: \[ T_k = \binom{2/3}{k} (3x)^k (-2)^{2/3 - k} \] Step 2: Find the fourth term, \( T_4 \), by setting \( k = 3 \) (since the term count starts from \( k = 0 \)).
- Compute: \[ T_4 = \binom{2/3}{3} \cdot 3^3 \cdot x^3 \cdot (-2)^{-1/3} \] - Simplify using properties of binomial coefficients and powers. 
Step 3: Evaluate the binomial coefficient and the negative exponent on \(-2\).
- \( \binom{2/3}{3} \) involves factorials and gamma functions for fractional parts, leading to: \[ T_4 = -\frac{\sqrt[3]{4}}{6} x^3 \]