How do you use a linear approximation or differentials to estimate $(8.06)^(⅔)$ ?

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Answer 1 · 2017-03-05T12:35:39

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$(8.06)^(2/3)$

$= (8 + 0.06)^(2/3)$

$= 8^(2/3)(1 + 0.06/8)^(2/3)$

$= 4(1 + 0.03/4)^(2/3)$

Binomial Series Expansion: $(1+ x)^alpha = 1 + alpha x + mathcal O (x^2)$

$= 4(1 + 2/3 * 0.03/4 + mathcal O (0.03/4 )^2)$

$approx 4(1 + 1/200) = 201/50 = 4.02$

How do you use a linear approximation or differentials to estimate (8.06)^(⅔)(8.06)^(⅔)?