Question

If \( x \) is numerically so small that \( x^{2} \) and higher powers of \( x \) can be neglected, then \( \left(1+\frac{2x}{3}\right)^{3/2} \cdot (32+5x)^{-1/5} \) is approximately equal to

• A. $\frac{32+31x}{64}$
• B. $\frac{31+32x}{64}$
• C. $\frac{31-32x}{64}$
• D. $\frac{1-2x}{64}$

Hint:

$(1+x)^n \approx 1 + nx$ when $|x| \ll 1$.

Answer 1

The Correct Option is A

Step 1: First Factor
Using $(1+y)^n \approx 1 + ny$ for small $y$: $(1+\frac{2x}{3})^{3/2} \approx 1 + \frac{3}{2}(\frac{2x}{3}) = 1 + x$.
Step 2: Second Factor
$(32+5x)^{-1/5} = (32)^{-1/5}(1+\frac{5x}{32})^{-1/5} = \frac{1}{2}(1 - \frac{1}{5}(\frac{5x}{32})) = \frac{1}{2}(1 - \frac{x}{32})$.
Step 3: Combining
$(1+x) \cdot \frac{1}{2}(1 - \frac{x}{32}) = \frac{1}{2}(1 + x - \frac{x}{32}) = \frac{1}{2}(1 + \frac{31x}{32}) = \frac{32+31x}{64}$.
Final Answer: (a)