Chapter 3.2, Problem 43E

Solution

To simplify the expression $x^{\frac{1}{4}}\cdot x^{\frac{1}{3}}$ .

  $x^{\frac{7}{12}}$

Given:

The expression $x^{\frac{1}{4}}\cdot x^{\frac{1}{3}}$ .

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 $x^{\frac{1}{4}}\cdot x^{\frac{1}{3}}$ , use the exponent properties and simplify further as shown below,

  $\begin{array}{c}{x}^{\frac{1}{4}}\cdot x^{\frac{1}{3}}=x^{\frac{1}{4}+\frac{1}{3}}\\ =x^{\frac{3}{12}+\frac{4}{12}}\\ =x^{\frac{3+4}{12}}\\ =x^{\frac{7}{12}}\end{array}$

Therefore, the simplified expression is $x^{\frac{7}{12}}$ .