Chapter 40, Problem 35A

To determine

(a)

Indicated root of the $\sqrt[7]{-1}$.

Answer to Problem 35A

Indicated root of the given number is, -1.

Explanation of Solution

Given:

  $\sqrt[7]{-1}$ is given as a signed number.

Calculation:

Here digit (-1) is expressed in seventh root.

  $\begin{array}{l}=\sqrt[7]{-1}\\ =\sqrt[7]{\left(-1\right)\times \left(-1\right)\times \left(-1\right)\times \left(-1\right)\times \left(-1\right)\times \left(-1\right)\times \left(-1\right)}\\ =\sqrt[7]{{(-1)}^7}\\ $ ={[(-1^7)]}^{\frac{1}{7}}$$ ={(-1)}^7{}^{\times \frac{1}{7}}$$ =-1$\end{array}$

Hence, the indicated root of the given number is, -1.

To determine

(b)

Indicated root of the $\sqrt[3]{216}$.

Answer to Problem 35A

Indicated root of the given number is, 6.

Explanation of Solution

Given:

  $\sqrt[3]{216}$ is given as a signed number.

Calculation:

Here digit (216) is expressed in cubic root.

  $\begin{array}{l}\sqrt[3]{216}\\ =\sqrt[3]{6\times 6\times 6}\\ =\sqrt[3]{{(6)}^3}\\ $ ={[(6^3)]}^{\frac{1}{3}}$$ ={(6)}^3{}^{\times \frac{1}{3}}$$ =6$\end{array}$

Hence, the indicated root of the given number is, 6.

To determine

(c)

Indicated root of the $\sqrt[3]{\frac{-8}{-64}}$.

Answer to Problem 35A

Indicated root of the given number is, $\frac{1}{2}$.

Explanation of Solution

Given:

  $\sqrt[3]{\frac{-8}{-64}}$ is given as a signed number.

Calculation:

Here digit $\frac{-8}{-64}$ is expressed in cubic root.

  $\begin{array}{l}=\sqrt[3]{\frac{-8}{-64}}\\ =\sqrt[3]{\frac{\left(-2\right)\times \left(-2\right)\times \left(-2\right)}{\left(-4\right)\times \left(-4\right)\times \left(-4\right)}}\\ =\sqrt[3]{\frac{{\left(-2\right)}^3}{{\left(-4\right)}^3}}\\ $ ={[{\left(\frac{-2}{-4}\right)}^3]}^{\frac{1}{3}}$$ ={\left(\frac{1}{2}\right)}^3{}^{\times \frac{1}{3}}$$ =\frac{1}{2}$\end{array}$

Hence, the indicated root of the given number is, $\frac{1}{2}$.

To determine

(d)

Indicated root of the $\sqrt[3]{\frac{+8}{-27}}$.

Answer to Problem 35A

Indicated root of the given number is, $\frac{+2}{-3}$.

Explanation of Solution

Given:

  $\sqrt[3]{\frac{+8}{-27}}$ is given as a signed number.

Calculation:

Here digit $\frac{+8}{-27}$ is expressed in cubic root.

  $\begin{array}{l}=\sqrt[3]{\frac{+8}{-27}}\\ =\sqrt[3]{\frac{\left(+2\right)\times \left(+2\right)\times \left(+2\right)}{\left(-3\right)\times \left(-3\right)\times \left(-3\right)}}\\ =\sqrt[3]{\frac{{(+2)}^3}{{\left(-3\right)}^3}}\\ ={[}^{{\left(\frac{+2}{-3}\right)}^3}\\ ={\left(\frac{+2}{-3}\right)}^3{}^{\times \frac{1}{3}}\\ =\frac{+2}{-3}\end{array}$

Hence, the indicated root of the given number is, $\frac{+2}{-3}$.

To determine

(e)

Indicated root of the $\sqrt[4]{\frac{+1}{+16}}$

Answer to Problem 35A

Indicated root of the given number is, $\frac{+1}{+2}$.

Explanation of Solution

Given:

  $\sqrt[4]{\frac{+1}{+16}}$ is given as a signed number.

Calculation:

Here digit $\frac{+1}{+16}$ is expressed in fourth root.

  $\begin{array}{l}=\sqrt[4]{\frac{+1}{+16}}\\ =\sqrt[4]{\frac{\left(+1\right)\times \left(+1\right)\times \left(+1\right)\times \left(+1\right)}{\left(+2\right)\times \left(+2\right)\times \left(+2\right)\times \left(+2\right)}}\\ =\sqrt[4]{\frac{{(+1)}^4}{{(+2)}^4}}\\ ={\left[{\left(\frac{+1}{+2}\right)}^4\right]}^{\frac{1}{4}}\\ ={\left(\frac{+1}{+2}\right)}^4{}^{\times \frac{1}{4}}\\ =\frac{+1}{+2}\end{array}$

Hence, the indicated root of the given number is, $\frac{+1}{+2}$.