Chapter 40, Problem 36A

To determine

(a)

Indicated root of the $\sqrt[3]{\frac{+27}{-125}}$.

Answer to Problem 36A

Indicated root of the given number is, $\frac{+3}{-5}$.

Explanation of Solution

Given:

  $\sqrt[3]{\frac{+27}{-125}}$ is given as a signed number.

Calculation:

Here, digit $\frac{+27}{-125}$ is expressed in cubic root.

  $\begin{array}{l}=\sqrt[3]{\frac{+27}{-125}}\\ =\sqrt[3]{\frac{\left(+3\right)\times \left(+3\right)\times \left(+3\right)}{\left(-5\right)\times \left(-5\right)\times \left(-5\right)}}\\ =\sqrt[3]{\frac{{(+3)}^3}{{\left(-5\right)}^3}}\\ $ ={[{\left(\frac{+3}{-5}\right)}^3]}^{\frac{1}{3}}$$ ={\left(\frac{+3}{-5}\right)}^3{}^{\times \frac{1}{3}}$$ =\frac{+3}{-5}$\end{array}$

Hence, the indicated root of the given number is, $\frac{+3}{-5}$.

To determine

(b)

Indicated root of the $\sqrt[3]{-236.539}$.

Answer to Problem 36A

Indicated root of the given number is, -6.1844.

Explanation of Solution

Given:

  $\sqrt[3]{-236.539}$ is given as a signed number.

Calculation:

Here, digit (-236.539) is expressed in cubic root.

  $\begin{array}{l}\sqrt[3]{-236.539}\\ =\sqrt[3]{\left(-6.1844\right)\times \left(-6.1844\right)\times \left(-6.1844\right)}\\ =\sqrt[3]{{\left(-6.1844\right)}^3}\\ $ ={[{\left(-6.1844\right)}^3]}^{\frac{1}{3}}$$ ={\left(-6.1844\right)}^3{}^{\times \frac{1}{3}}$$ =-6.1844$\end{array}$

Hence, the indicated root of the given number is, -6.1844.

To determine

(c)

Indicated root of the $\sqrt[5]{-86.009}$.

Answer to Problem 36A

Indicated root of the given number is, -2.4373.

Explanation of Solution

Given:

  $\sqrt[5]{-86.009}$ is given as a signed number.

Calculation:

Here, digit (-86.009) is expressed in fifth root.

  $\begin{array}{l}\sqrt[5]{-86.009}\\ =\sqrt[5]{\left(-2.4373\right)\times \left(-2.4373\right)\times \left(-2.4373\right)\times \left(-2.4373\right)\times \left(-2.4373\right)}\\ =\sqrt[5]{{\left(-2.4373\right)}^5}\\ $ ={[{\left(-2.4373\right)}^5]}^{\frac{1}{5}}$$ ={\left(-2.4373\right)}^5{}^{\times \frac{1}{5}}$$ =-2.4373$\end{array}$

Hence, the indicated root of the given number is, -2.4373.

To determine

(d)

Indicated root of the $\sqrt[3]{\frac{-97.326}{+123.592}}$.

Answer to Problem 36A

Indicated root of the given number is, $\frac{-4.5998}{+5.9812}$.

Explanation of Solution

Given:

  $\sqrt[3]{\frac{-97.326}{+123.592}}$ is given as a signed number.

Calculation:

Here, digit $\frac{-97.326}{123.592}$ is expressed in cubic root.

  $\begin{array}{l}=\sqrt[3]{\frac{-97.326}{+123.592}}\\ =\sqrt[3]{\frac{\left(-4.5998\right)\times \left(-4.5998\right)\times \left(-4.5998\right)}{\left(+5.9812\right)\times \left(+5.9812\right)\times \left(+5.9812\right)}}\\ =\sqrt[3]{\frac{{\left(-4.5998\right)}^3}{{\left(+5.9812\right)}^3}}\\ $ ={[{\left(\frac{-4.5998}{+5.9812}\right)}^3]}^{\frac{1}{3}}$$ ={\left(\frac{-4.5998}{+5.9812}\right)}^3{}^{\times \frac{1}{3}}$$ =\frac{-4.5998}{+5.9812}$\end{array}$

Hence, the indicated root of the given number is, $\frac{-4.5998}{+5.9812}$.

To determine

(e)

Indicated root of the $\sqrt[3]{\frac{-89.096}{-17.323}}$.

Answer to Problem 36A

Indicated root of the given number is, $\frac{-4.4664}{-2.5875}$.

Explanation of Solution

Given:

  $\sqrt[3]{\frac{-89.096}{-17.323}}$ is given as a signed number.

Calculation:

Here, digit $\frac{-89.096}{-17.323}$ is expressed in cubic root.

  $\begin{array}{l}=\sqrt[3]{\frac{-89.096}{-17.323}}\\ =\sqrt[3]{\frac{\left(-4.4664\right)\times \left(-4.4664\right)\times \left(-4.4664\right)}{\left(-2.5875\right)\times \left(-2.5875\right)\times \left(-2.5875\right)}}\\ =\sqrt[3]{\frac{{\left(-4.4664\right)}^3}{{\left(-2.5875\right)}^3}}\\ $ ={[{\left(\frac{-4.4664}{-2.5875}\right)}^3]}^{\frac{1}{3}}$$ ={\left(\frac{-4.4664}{-2.5875}\right)}^3{}^{\times \frac{1}{3}}$$ =\frac{-4.4664}{-2.5875}$\end{array}$

Hence, the indicated root of the given number is, $\frac{-4.4664}{-2.5875}$.