Chapter 3.2, Problem 42E

Solution

To describe and correct the error in simplifying the given expression.

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

Given:

The expression,

  

Concept Used:

• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$
• Product of power property: $a^m\cdot a^n=a^{m+n}$
• Quotient of power property: $\frac{a^m}{a^n}=a^{m-n}$
• Product property of radicals: $\sqrt[n]{a}\cdot \sqrt[n]{b}=\sqrt[n]{a\cdot b}$
• Quotient property of radicals: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• Sum of like radicals’ property: $p\sqrt[n]{a}+q\sqrt[n]{a}=\left(p+q\right)\sqrt[n]{a}$
• The exponent form of the nth root of a number a is given by $\sqrt[n]{a}=a^{\frac{1}{n}}$

Calculation:

Recall that to rationalize the denominator, the same variable must be multiplied with numerator and denominator both.

Note that in the second term, the variable y is only multiplied with the denominator, which is wrong.

Thus, the corrected expression is,

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