Chapter 7.5, Problem 2P

a.

To determine

To determine each exponential expression in radical form.

Answer to Problem 2P

  $a\sqrt[5]{a}$

Explanation of Solution

Given:

  $a^{\frac{5}{6}}$

Formula used:

  $x^{\frac{1}{n}}=\sqrt[n]{x}$

Calculation:

The given expression is,

  $\Rightarrow a^{\frac{5}{6}}$

Breaking it into radical form,

  $\Rightarrow a^{5\times \frac{1}{6}}$

Now, if in power fraction is there then it can be transformed into root,

  $\Rightarrow \sqrt[6]{a^5}$$\left[\because x^{\frac{1}{n}}=\sqrt[n]{x}\right]$

  $\Rightarrow \sqrt[5]{a\times a\times a\times a\times a\times a}$

Now,

  $\Rightarrow \sqrt[5]{a\times a\times a\times a\times a}\times \sqrt[5]{a}$

  $\Rightarrow a\times \sqrt[5]{a}$

Conclusion:

Therefore, the radial form of $a^{\frac{5}{6}}$ is $a\sqrt[5]{a}$ .

b.

To determine

To determine each exponential expression in radical form.

Answer to Problem 2P

  $5\sqrt[3]{x}$

Explanation of Solution

Given:

  $5x^{\frac{1}{3}}$

Formula used:

  $x^{\frac{1}{n}}=\sqrt[n]{x}$

Calculation:

The given expression is,

  $\Rightarrow 5x^{\frac{1}{3}}$

Breaking it into radical form,

  $\Rightarrow 5x^{\frac{1}{3}}$

Now, if in power fraction is there then it can be transformed into root,

  $\Rightarrow 5\times \sqrt[3]{x}$$\left[\because x^{\frac{1}{n}}=\sqrt[n]{x}\right]$

  $\Rightarrow 5\sqrt[3]{x}$

Conclusion:

Therefore, the radial form of $5x^{\frac{1}{3}}$ is $5\sqrt[3]{x}$ .

c.

To determine

To determine each exponential expression in radical form.

Answer to Problem 2P

  $9\sqrt[3]{4x^2}$

Explanation of Solution

Given:

  ${\left(54y\right)}^{\frac{2}{3}}$

Formula used:

  $x^{\frac{1}{n}}=\sqrt[n]{x}$

Calculation:

The given expression is,

  $\Rightarrow {\left(54y\right)}^{\frac{2}{3}}$

Breaking it into radical form,

  $\Rightarrow {\left(54y\right)}^{\frac{2}{3}}$

  $\Rightarrow {\left[{\left(54y\right)}^2\right]}^{\frac{1}{3}}$

  $\Rightarrow {\left[\left(2916x^2\right)\right]}^{\frac{1}{3}}$

Now, if in power fraction is there then it can be transformed into root,

  $\Rightarrow 9\times \sqrt[3]{4x^2}$$\left[\because x^{\frac{1}{n}}=\sqrt[n]{x}\right]$

  $\Rightarrow 9\sqrt[3]{4x^2}$

Conclusion:

Therefore, the radial form of ${\left(54y\right)}^{\frac{2}{3}}$ is $9\sqrt[3]{4x^2}$ .