Chapter 10.6, Problem 8WE

To determine

To Find:

  $\sqrt[3]{{\left(412\right)}^2}$

Answer to Problem 8WE

  $\sqrt[3]{{\left(412\right)}^2}=55.369$

Explanation of Solution

Given:

  $\sqrt[3]{{\left(412\right)}^2}$

Formula Used:

The log table:

  

Calculation:

Let

  $\begin{array}{l}\sqrt[3]{{\left(412\right)}^2}=x\\ \Rightarrow {\left(412\right)}^{\frac{2}{3}}=x\\ \Rightarrow {\log }_{10}{\left(412\right)}^{\frac{2}{3}}={\log }_{10}x\mathrm{........}[\text{take log each side]}\\ \Rightarrow \frac{2}{3}\log 412=\log x\end{array}$

We evaluate

  $\begin{array}{l}log412\\ =\log 4.12\times {10}^2\\ =\log 4.12+{\log }_{10}{10}^2\mathrm{..........}[{\log }_{b}MN={\log }_{b}M+{\log }_{b}N]\\ =\log 4.12+\mathrm{2.....................}[{\log }_{b}b=1;{\log }_b{M}^k=k{\log }_{b}M]\end{array}$

We evaluate log4.12:

1. The characteristic part is 4 and mantissa part is 12.
2. Check the row number 41 and column number 2. So, the value obtained is 6149, which is the mantissa part of the solution.
3. Since the characteristic part is 4 , which is in between ${10}^0\le 4\le {10}^1$ , so the characteristic part of the solution is 0.
4. Combine the characteristic part and the mantissa part , we get, log4.12 = 0.6149.
5. So, log412 = 0.6149+2 = 2.6149

So,

  $\begin{array}{l}\frac{2}{3}\log 412=\log x\\ \Rightarrow \log x=\frac{2}{3}(2.6149)\\ \Rightarrow \log x=1.7432667\\ \Rightarrow {10}^{{\log }_{10}x}={10}^{1.7432667}\mathrm{...............}[\text{raise 10 to the power each side of the equation]}\\ \Rightarrow x=\mathrm{55.369.......................}[b^{{\log }_{b}x}=x]\end{array}$

So, the solution is $\sqrt[3]{{\left(412\right)}^2}=55.369$