Chapter 3.2, Problem 10Q

Solution

To simplify the expression ${\left(\sqrt{10}\cdot \sqrt[3]{10}\right)}^8$ .

  ${10}^{\frac{20}{3}}$

Given:

The expression ${\left(\sqrt{10}\cdot \sqrt[3]{10}\right)}^8$ .

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:

In order to simplify the given expression ${\left(\sqrt{10}\cdot \sqrt[3]{10}\right)}^8$ , use the exponent properties and simplify further as shown below,

  $\begin{array}{c}{\left(\sqrt{10}\cdot \sqrt[3]{10}\right)}^8={\left({10}^{\frac{1}{2}}\cdot {10}^{\frac{1}{3}}\right)}^8\\ ={\left({10}^{\frac{1}{2}+\frac{1}{3}}\right)}^8\\ ={\left({10}^{\frac{3+2}{6}}\right)}^8\\ ={\left({10}^{\frac{5}{6}}\right)}^8\\ ={10}^{\frac{5}{6}\cdot 8}\\ ={10}^{\frac{20}{3}}\end{array}$

Therefore, the simplified expression is ${10}^{\frac{20}{3}}$ .