Chapter 3.2, Problem 18E

Solution

To simplify the expression ${\left(\sqrt[3]{3}\cdot \sqrt[4]{3}\right)}^{12}$ .

  $2187$

Given:

The expression ${\left(\sqrt[3]{3}\cdot \sqrt[4]{3}\right)}^{12}$ .

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: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• 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 ${\left(\sqrt[3]{3}\cdot \sqrt[4]{3}\right)}^{12}$ , use the exponent properties and simplify further as shown below,

  $\begin{array}{c}{\left(\sqrt[3]{3}\cdot \sqrt[4]{3}\right)}^{12}={\left(3^{\frac{1}{3}}\cdot 3^{\frac{1}{4}}\right)}^{12}\\ =3^{\frac{1}{3}\cdot 12}\cdot 3^{\frac{1}{4}\cdot 12}\\ =3^4\cdot 3^3\\ =3^{4+3}\\ =3^7\\ =2187\end{array}$

Therefore, the simplified expression is $2187$ .