Chapter 3.2, Problem 41E

Solution

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

  $\begin{array}{c}2\sqrt[3]{10}+6\sqrt[3]{5}=2\sqrt[3]{2\cdot 5}+6\sqrt[3]{5}\\ =2\sqrt[3]{2}\cdot \sqrt[3]{5}+6\sqrt[3]{5}\\ =\left(2\sqrt[3]{2}+6\right)\sqrt[3]{5}\end{array}$

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 two radical terms can be added or subtracted only when the order of the radical are same and the expression inside the radical are same as well.

Note that the number inside the cube root on the left side are not equal, so the given expression cannot be simplified further.

Thus, the corrected expression is,

  $\begin{array}{c}2\sqrt[3]{10}+6\sqrt[3]{5}=2\sqrt[3]{2\cdot 5}+6\sqrt[3]{5}\\ =2\sqrt[3]{2}\cdot \sqrt[3]{5}+6\sqrt[3]{5}\\ =\left(2\sqrt[3]{2}+6\right)\sqrt[3]{5}\end{array}$