First slide

Binomial theorem for negative integral and rational Index

Question

$If x is so small, higher powers of x may be neglected then \sqrt[3]{x^{2} + 27} - \sqrt[3]{x^{2} + 8} =$

Moderate

A

$2 - \frac{5 x^{2}}{24}$
B

$1 + \frac{5 x^{2}}{108}$
C

$1 - \frac{5 x^{2}}{108}$
D

$2 + \frac{5 x^{2}}{24}$

Solution

$\begin{array}{l} \sqrt[3]{x^{2} + 27} - \sqrt[3]{x^{2} + 8} = 3 \sqrt[3]{[1 + \frac{x^{2}}{27}]} - 2 \sqrt[3]{[1 + \frac{x^{2}}{8}}] \\ = 3 {(1 + \frac{x^{2}}{27})}^{1 / 3} - 2 {(1 + \frac{x^{2}}{8})}^{1 / 3} \\ = 3 (1 + \frac{x^{2}}{81}) - 2 (1 + \frac{x^{2}}{24}) == 3 + \frac{x^{2}}{27} - 2 - \frac{x^{2}}{12} \\ = 1 - \frac{5 x^{2}}{108} . \end{array}$

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