If x is numerically so small that x^2 and hig | vedclass

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Question

(A) Given expression: $E = (1 + \frac{2x}{3})^{3/2} \cdot (32 + 5x)^{-1/5}$ Using the binomial approximation $(1+u)^n \approx 1+nu$ for small $u$: $E

Vedclass · Accepted Answer

$\frac{32+31x}{64}$

Vedclass · Answer

(A) Given expression: $E = (1 + \frac{2x}{3})^{3/2} \cdot (32 + 5x)^{-1/5}$ Using the binomial approximation $(1+u)^n \approx 1+nu$ for small $u$: $E \approx (1 + \frac{3}{2} \cdot \frac{2x}{3}) \cdot (32)^{-1/5} (1 + \frac{5x}{32})^{-1/5}$ $E \approx (1 + x) \cdot \frac{1}{2} \cdot (1 - \frac{1}{5} \cdot \frac{5x}{32})$ $E \approx \frac{1}{2} (1 + x) (1 - \frac{x}{32})$ Neglecting $x^2$ terms: $E \approx \frac{1}{2} (1 + x - \frac{x}{32}) = \frac{1}{2} (1 + \frac{31x}{32}) = \frac{32 + 31x}{64}$

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

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