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how do you calculate the cube root without calculator

Guest May 29, 2014

Best Answer 

 #3
avatar+26407 
+5

The iterative method Bertie mentioned, to find the cube root of "a", say, is:

$$$$x_{n+1}=\frac{1}{3}(\frac{a}{x_n^2}+2x_n)$$

This is derived using Newton-Raphson.

However, in general, I wouldn't want to use this without a calculator of some sort! 

Alan  May 31, 2014
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3+0 Answers

 #2
avatar+889 
+5

There is, I vaguely remember, an algebraic method similar to that for calculating square roots and it's also possible to use the Newton-Raphson method, (though you would need to be good at arithmetic, hardly the sort of calculation you would attempt without a calculator).

If the number is close to some convenient cube, it's possible to make use of the binomial expansion. For example.

$$\begin{array}{lcl}
\sqrt[3]{1005}&=&(1000+5)^{1/3}\\
&=&1000^{1/3}(1+0.005)^{1/3}\\
&=&10(1+(1/3)0.005+(1/3)(-2/3)0.005^{2}/2!+\dots)\\
&=&10(1+0.0016667-0.0000028+\dots)\\
&\approx&10.016639
\end{array}$$

That's easily done on paper and is correct to 6dp.

Bertie  May 30, 2014
 #3
avatar+26407 
+5
Best Answer

The iterative method Bertie mentioned, to find the cube root of "a", say, is:

$$$$x_{n+1}=\frac{1}{3}(\frac{a}{x_n^2}+2x_n)$$

This is derived using Newton-Raphson.

However, in general, I wouldn't want to use this without a calculator of some sort! 

Alan  May 31, 2014
 #4
avatar+81077 
0

That's an interesting tecnique, Alan and Bertie......

I'm with Alan...........a calculator seems vital......

 

CPhill  May 31, 2014

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