Chapter 10.6, Problem 6WE

To determine

To Find:

  $\sqrt[6]{786}$

Answer to Problem 6WE

  $\sqrt[6]{786}=3.038$

Explanation of Solution

Given:

  $\sqrt[6]{786}$

Formula Used:

The log table:

  

Calculation:

Let

  $\begin{array}{l}\sqrt[6]{786}=x\\ \Rightarrow {\log }_{10}{786}^{\frac{1}{6}}={\log }_{10}x\mathrm{........}[\text{take log each side]}\\ \Rightarrow \frac{1}{6}\log 786=\log x\end{array}$

We evaluate

  $\begin{array}{l}log786\\ =\log 7.86\times {10}^2\\ =\log 7.86+{\log }_{10}{10}^2\mathrm{..........}[{\log }_{b}MN={\log }_{b}M+{\log }_{b}N]\\ =\log 7.86+\mathrm{2.....................}[{\log }_{b}b=1;{\log }_b{M}^k=k{\log }_{b}M]\end{array}$

We evaluate log7.86:

1. The characteristic part is 7 and mantissa part is 86.
2. Check the row number 78 and column number 6. So, the value obtained is 8954, which is the mantissa part of the solution.
3. Since the characteristic part is 7 , which is in between ${10}^0\le 7\le {10}^1$ , so the characteristic part of the solution is 0.
4. Combine the characteristic part and the mantissa part , we get, log7.86 = 0.8954.
5. So, log786 = 0.8954+2 = 2.8954

So,

  $\begin{array}{l}\frac{1}{6}\log 786=\log x\\ \Rightarrow \log x=\frac{1}{6}(2.8954)\\ \Rightarrow \log x=0.48256\\ \Rightarrow {10}^{{\log }_{10}x}={10}^{0.48256}\mathrm{...............}[\text{raise 10 to the power each side of the equation]}\\ \Rightarrow x=\mathrm{3.038.......................}[b^{{\log }_{b}x}=x]\end{array}$

So, the solution is $\sqrt[6]{786}=3.038$