Binomial series - Finding square root of number problem

[stfz]

Homework Statement

Expand $(1+x)^(1/3)$ in ascending powers of x as far as the term $x^3$, simplifying the terms as much as possible. By substituting 0.08 for x in your result, obtain an approximate value of the cube root of 5, giving your answer to four places of decimals.

Homework Equations

Binomial series.

The Attempt at a Solution

Expansion as per binomial series :
$1+(1/3)x - (1/9)x^2 + (5/81)x^3$.

No idea how to find cube root of 5 by using x = 0.08.
Help please!
Sorry if I am asking too many questions - I don't have a tutor for this, I'm on my own

Thanks! :)

 

[HallsofIvy]

Are you sure you have read the problem correctly? Putting x= 0.08 into that formula will give you $(0.08+ 1)^{1/3}= 1.08^{1/3}$, not $\sqrt[3]{5}$. 5= 4+ 1 so you should set x= 4 to use that formula to find $\sqrt[3]{5}$.

 

[Ray Vickson]

[

Are you sure you have read the problem correctly? Putting x= 0.08 into that formula will give you $(0.08+ 1)^{1/3}= 1.08^{1/3}$, not $\sqrt[3]{5}$. 5= 4+ 1 so you should set x= 4 to use that formula to find $\sqrt[3]{5}$.

That would approximate $\sqrt[3]{5}$ by a few terms of a divergent series, which does not look like a good idea. However, using $2^3 = 8$ we can write
$$5 = 8 - 3 = 8 \left(1 - \frac{3}{8}\right) \Longrightarrow 5^{1/3} = 2 (1 - 0.375)^{1/3}$$

BTW: please try not to confuse everybody by using a bad thread title. Don't say "square root" when you mean "cube root".