# What is the cube root of 47.1 ?

Mar 21, 2017

$\sqrt[3]{47.1} \approx 3.6113837181884034$

#### Explanation:

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

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

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

$\sqrt[3]{47.1} = {e}^{\frac{1}{3} \ln \left(47.1\right)} \approx 3.61138372$

Otherwise, we can find rational approximations as follows:

Given an approximation ${a}_{0}$, iterate using the formula:

${a}_{i + 1} = {a}_{i} + \frac{47.1 - {a}_{i}^{3}}{3 {a}_{i}^{2}}$

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

${a}_{0} = 3.5$

${a}_{1} \approx 3.6149659863945578$

${a}_{2} \approx 3.6113872668819988$

${a}_{3} \approx 3.6113837181918909$

${a}_{4} \approx 3.6113837181884034$

${a}_{5} \approx 3.6113837181884034$

As you can see, this converges quite rapidly.