Chapter 3.2, Problem 80E

Solution

The simplified form of the expression ${\left(\frac{x^{\pi }}{x^{\frac{\pi }{3}}}\right)}^2$ .

  $x^{\frac{4\pi }{3}}$

Given:

The expression ${\left(\frac{x^{\pi }}{x^{\frac{\pi }{3}}}\right)}^2$ .

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 ${\left(\frac{x^{\pi }}{x^{\frac{\pi }{3}}}\right)}^2$ , use the exponent properties and combine the like terms to simplify further as shown below,

  $\begin{array}{c}{\left(\frac{x^{\pi }}{x^{\frac{\pi }{3}}}\right)}^2=\frac{x^{\pi \cdot 2}}{x^{\frac{\pi }{3}\cdot 2}}\\ =\frac{x^{2\pi }}{x^{\frac{2\pi }{3}}}\\ =x^{2\pi -\frac{2\pi }{3}}\\ =x^{\frac{6\pi }{3}-\frac{2\pi }{3}}\\ =x^{\frac{6\pi -2\pi }{3}}\\ =x^{\frac{4\pi }{3}}\end{array}$

Therefore, the simplified expression is $x^{\frac{4\pi }{3}}$ .