Chapter 40, Problem 38A

To determine

(a)

The value of equation $-{125}^{\frac{1}{3}}.$

Answer to Problem 38A

The value of given equation is -5.

Explanation of Solution

Given:

Equation is given in problem.

Calculation:

  $\begin{array}{l}=-{125}^{\frac{1}{3}}\\ =-\sqrt[3]{125}\\ =-\sqrt[3]{5\times 5\times 5}\\ =-\sqrt[3]{{(5)}^3}\\ $ ={[{(5)}^3]}^{\frac{1}{3}}$$ ={(5)}^3{{}^{\times }}^{\frac{1}{3}}$$ =5$\end{array}$

To determine

(b)

The value of equation ${125}^{\frac{1}{3}}.$

Answer to Problem 38A

The value of given equation is5.

Explanation of Solution

Given:

Use the proper order of operation.

Calculation:

  $\begin{array}{l}=-{125}^{\frac{1}{3}}\\ =-\sqrt[3]{125}\\ =-\sqrt[3]{5\times 5\times 5}\\ =-\sqrt[3]{{(5)}^3}\\ $ ={[{(5)}^3]}^{\frac{1}{3}}$$ ={(5)}^3{{}^{\times }}^{\frac{1}{3}}$$ =5$\end{array}$

To determine

(c)

The value of equation ${273.19}^{\frac{2}{3}}.$

Answer to Problem 38A

The value of given equation is $42.10$.

Explanation of Solution

Given:

Use the proper order of operation.

Calculation:

  $={(273.19)}^{\frac{2}{3}}$

  $=\sqrt[3]{{(273.19)}^2}$

  $=\sqrt[3]{273.19\times 273.19}$

  $=\sqrt[3]{74632.78}$

  $=\sqrt[3]{(42.10\times 42.10\times 42.10)}$

  $=\sqrt[3]{{(42.10)}^3}$

  $={[{(42.10)}^3]}^{\frac{1}{3}}$

  $={(42.10)}^{3\times \frac{1}{3}}$

  $=42.10$

To determine

(d)

The value of equation ${41.673}^{\frac{-1}{2}}.$

Answer to Problem 38A

The value of given equation is $\mathrm{0.155.}$

Explanation of Solution

Given:

Use the proper order of operation.

Calculation:

  $\begin{array}{l}={41.673}^{\frac{-1}{2}}\\ =\frac{1}{{41.673}^{\frac{1}{2}}}\\ =\frac{1}{\sqrt{41.673}}\\ =\frac{1}{6.455}\\ =0.155\end{array}$

To determine

(e)

The value of equation ${8.007}^{\frac{2}{3}}.$

Answer to Problem 38A

The value of given equation is $\mathrm{4.002.}$

Explanation of Solution

Given:

Use the proper order of operation.

Calculation:

  $={(8.007)}^{\frac{2}{3}}$

  $=\sqrt[3]{{(8.007)}^2}$

  $=\sqrt[3]{8.007\times 8.007}$

  $=\sqrt[3]{64.112}$

  $=\sqrt[3]{(4.002\times 4.002\times 4.002)}$

  $=\sqrt[3]{{(4.002)}^3}$

  $={[{(4.002)}^3]}^{\frac{1}{3}}$

  $={(4.002)}^{3\times \frac{1}{3}}$

  $=4.002$

To determine

(f)

The value of equation ${67.725}^{\frac{-2}{3}}.$

Answer to Problem 38A

The value of given equation is $\mathrm{0.06.}$

Explanation of Solution

Given:

Use the proper order of operation.

Calculation:

  $\begin{array}{l}={67.725}^{\frac{-2}{3}}\\ =\frac{1}{{67.725}^{\frac{2}{3}}}\\ =\frac{1}{\sqrt[3]{{(67.725)}^2}}\\ =\frac{1}{\sqrt[3]{(67.725\times 67.725)}}\\ =\frac{1}{\sqrt[3]{(4586.676)}}\\ =\frac{1}{\sqrt[3]{(16.615\times 16.615\times 16.615)}}\\ =\frac{1}{\sqrt[3]{{(16.615)}^3}}\\ =\frac{1}{{[{(16.615)}^3]}^{\frac{1}{3}}}\\ =\frac{1}{{(16.615)}^3{}^{\times \frac{1}{3}}}\\ =\frac{1}{16.615}\\ =0.06\\ \end{array}$