Chapter 7.5, Problem 50PPE

(a)

To determine

To show: that $\sqrt{x^2}=x$ by rewriting $\sqrt{x^2}$ in exponential form

Answer to Problem 50PPE

Hence Proved that the exponential form of $\sqrt{x^2}$ is $x$

Explanation of Solution

Given info:

Show that $\sqrt{x^2}=x$ by rewriting $\sqrt{x^2}$ in exponential form

Formula used:

Show the rational exponential form as follows:

  $\begin{array}{l}\sqrt{x}=x^{\frac{1}{2}}\\ \sqrt[n]{x}=x^{\frac{1}{n}}\\ \sqrt[n]{x^2}={\left(x^2\right)}^{\frac{1}{n}}=x^{\frac{2}{n}}\end{array}$

Here, $x$ is the variable, and the denominator in the power of $x$ represents the rational exponent.

Calculation:

Find the exponential form of $\sqrt{x^2}$ as follows:

  $\begin{array}{l}\sqrt{x^2}\\ {\left(x^2\right)}^{\frac{1}{2}}\\ x^{\frac{2}{2}}\\ x^1\\ x\end{array}$

Hence, Proved that the exponential form of $\sqrt{x^2}$ is $x$

Conclusion:

Thus, Proved that the exponential form of $\sqrt{x^2}$ is $x$ .

(b)

To determine

To show: that $\sqrt[4]{x^2}=\sqrt{x}$ by rewriting $\sqrt[4]{x^2}$ in exponential form

Answer to Problem 50PPE

Hence proved that the exponential form $\sqrt[4]{x^2}$ is $\sqrt{x}$

Explanation of Solution

Given info:

Show that $\sqrt[4]{x^2}=\sqrt{x}$ by rewriting $\sqrt[4]{x^2}$ in exponential form

Formula used:

Show the rational exponential form as follows:

  $\begin{array}{l}\sqrt{x}=x^{\frac{1}{2}}\\ \sqrt[n]{x}=x^{\frac{1}{n}}\\ \sqrt[n]{x^2}={\left(x^2\right)}^{\frac{1}{n}}=x^{\frac{2}{n}}\end{array}$

Here, $x$ is the variable, and the denominator in the power of $x$ represents the rational exponent.

Calculation:

Find the exponential form of $\sqrt[4]{x^2}$ as follows:

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

Hence, Proved that the exponential form of $\sqrt[4]{x^2}$ is $\sqrt{x}$

Conclusion:

Thus, Proved that the exponential form of $\sqrt[4]{x^2}$ is $\sqrt{x}$