What is the cube root of $47.1$ $47.1$ ?

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What is the cube root of $47.1$ $47.1$ ?

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Answer 1 · 2017-03-21T22:25:02

$\sqrt[3]{47.1} \approx 3.6113837181884034$ $root(3)(47.1) ~~ 3.6113837181884034$

Explanation:

$\sqrt[3]{47.1}$ $root(3)(47.1)$ is an irrational number somewhere between $3$ $3$ and $4$ $4$ , since:

$3^{3} = 27 < 47.1 < 64 = 4^{3}$ $3^3 = 27 < 47.1 < 64 = 4^3$

If you have a calculator with $e^{x}$ $e^x$ and $ln x$ $ln x$ functions, then you can calculate an approximation using:

$\sqrt[3]{47.1} = e^{\frac{1}{3} ln (47.1)} \approx 3.61138372$ $root(3)(47.1) = e^(1/3 ln(47.1)) ~~ 3.61138372$

Otherwise, we can find rational approximations as follows:

Given an approximation $a_{0}$ $a_0$ , iterate using the formula:

$a_{i + 1} = a_{i} + \frac{47.1 - a_{i}^{3}}{3 a_{i}^{2}}$ $a_(i+1) = a_i+(47.1 - a_i^3)/(3a_i^2)$

Putting $a_{0} = 3.5$ $a_0 = 3.5$ into a spreadsheet, I got the following sequence of approximations:

$a_{0} = 3.5$ $a_0 = 3.5$

$a_{1} \approx 3.6149659863945578$ $a_1 ~~ 3.6149659863945578$

$a_{2} \approx 3.6113872668819988$ $a_2 ~~ 3.6113872668819988$

$a_{3} \approx 3.6113837181918909$ $a_3 ~~ 3.6113837181918909$

$a_{4} \approx 3.6113837181884034$ $a_4 ~~ 3.6113837181884034$

$a_{5} \approx 3.6113837181884034$ $a_5 ~~ 3.6113837181884034$

As you can see, this converges quite rapidly.

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What is the cube root of $47.1$ $47.1$ ?

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