Chapter 10.6, Problem 7WE

To determine

To Find:

  $\sqrt[5]{{\left(81.2\right)}^4}$

Answer to Problem 7WE

  $\sqrt[5]{{\left(81.2\right)}^4}=33.704$

Explanation of Solution

Given:

  $\sqrt[5]{{\left(81.2\right)}^4}$

Formula Used:

The log table:

  

Calculation:

Let

  $\begin{array}{l}\sqrt[5]{{\left(81.2\right)}^4}=x\\ \Rightarrow {\left(81.2\right)}^{\frac{4}{5}}=x\\ \Rightarrow {\log }_{10}{\left(81.2\right)}^{\frac{4}{5}}={\log }_{10}x\mathrm{........}[\text{take log each side]}\\ \Rightarrow \frac{4}{5}\log 81.2=\log x\end{array}$

We evaluate

  $\begin{array}{l}log81.2\\ =\log 8.12\times 10\\ =\log 8.12+{\log }_{10}\mathrm{10..........}[{\log }_{b}MN={\log }_{b}M+{\log }_{b}N]\\ =\log 8.12+\mathrm{1.....................}[{\log }_{b}b=1;{\log }_b{M}^k=k{\log }_{b}M]\end{array}$

We evaluate log8.12:

1. The characteristic part is 8 and mantissa part is 12.
2. Check the row number 81 and column number 2. So, the value obtained is 9096, which is the mantissa part of the solution.
3. Since the characteristic part is 8 , which is in between ${10}^0\le 8\le {10}^1$ , so the characteristic part of the solution is 0.
4. Combine the characteristic part and the mantissa part , we get, log8.12 = 0.9096.
5. So, log81.2 = 0.9096+1 = 1.9096

So,

  $\begin{array}{l}\frac{4}{5}\log 81.2=\log x\\ \Rightarrow \log x=\frac{4}{5}(1.9096)\\ \Rightarrow \log x=1.52768\\ \Rightarrow {10}^{{\log }_{10}x}={10}^{1.52768}\mathrm{...............}[\text{raise 10 to the power each side of the equation]}\\ \Rightarrow x=\mathrm{33.704.......................}[b^{{\log }_{b}x}=x]\end{array}$

So, the solution is $\sqrt[5]{{\left(81.2\right)}^4}=33.704$