Solved: Write down and simplify the first three terms, in ascending powers of $x$, in the... [IB Mathematics Analysis and Approaches (AA)]

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HLPaper 2

Write down and simplify the first three terms, in ascending powers of $x$ , in the Extended Binomial expansion of $(1 - x)^{\frac{1}{3}}$ .

[3]

Verified

Solution

${(1 - x)^{\frac{1}{3}}} = 1 + \frac{1}{3}(-x) + \frac{1}{3}\left(\frac{{-2}}{3}\right)\frac{{(-x)^2}}{{2!}} \ldots = 1 - \frac{x}{3} - \frac{{x^2}}{9} \ldots$ M1A1A1

By substituting $x = \frac{1}{9}$ find a rational approximation to $\sqrt[3]{9}$ .

[3]

Verified

Solution

${\left( {\frac{8}{9}} \right)^{\frac{1}{3}}} \simeq 1 - \frac{1}{{27}} - \frac{1}{{729}} \Rightarrow \frac{2}{{\sqrt[3]{9}}} \simeq \frac{{701}}{{729}} \Rightarrow \sqrt[3]{9} \simeq \frac{{1458}}{{701}}$ M1A1A1