Chapter 6.2, Problem 55E

To determine

To verify: if the given statement is always, sometimes or never true.

Answer to Problem 55E

Sometimes true

Explanation of Solution

Given information:

The statement, “for a nonnegative real number x , $\frac{x^{\frac{2}{3}}}{x^{\frac{1}{3}}}=\sqrt[3]{x}$ ”

Concept used:

Quotient of power property:

  $\frac{a^m}{a^n}=a^{m-n}$

Rational Exponent: for any integer $n>1$

  $\sqrt[n]{a}=a^{\frac{1}{n}}$

By the above definition of rational exponent, taking $n=3$ , we have

  $\sqrt[3]{x}=x^{\frac{1}{3}}$

Also, by Quotient of power property, for $x\ne 0$

  $\begin{array}{c}\frac{x^{\frac{2}{3}}}{x^{\frac{1}{3}}}=x^{\frac{2}{3}-\frac{1}{3}}\\ =x^{\frac{1}{3}}\end{array}$

Thus, for $x\ne 0$

  $\begin{array}{c}\frac{x^{\frac{2}{3}}}{x^{\frac{1}{3}}}=x^{\frac{1}{3}}\\ =\sqrt[3]{x}\end{array}$

For $x=0$ , the statement $\frac{x^{\frac{2}{3}}}{x^{\frac{1}{3}}}=\sqrt[3]{x}$ is not true because division by 0 is undefined.

So, the given statement is sometimes true.