Chapter 3.2, Problem 51E

Solution

To write two variable expressions with non-integer exponents whose quotient is $x^{\frac{3}{4}}$ .

  $x^{\frac{5}{4}}\text{ and }{x}^{\frac{1}{2}}$

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: $\frac{\sqrt[n]{a}}{\sqrt[n]{b}}=\sqrt[n]{\frac{a}{b}}$
• 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 find such expressions with non-integer exponents whose quotient is $x^{\frac{3}{4}}$ , find two such fractions whose difference is $\frac{3}{4}$ .

Note that two such fractions whose difference is $\frac{3}{4}$ are $\frac{5}{4}$ , and $\frac{1}{2}$ .

Hence two such expressions are $x^{\frac{5}{4}}\text{ and }{x}^{\frac{1}{2}}$ , because the quotient is

  $\begin{array}{c}\frac{x^{\frac{5}{4}}}{x^{\frac{1}{2}}}=x^{\frac{5}{4}-\frac{1}{2}}\\ =x^{\frac{5}{4}-\frac{2}{4}}\\ =x^{\frac{5-2}{4}}\\ =x^{\frac{3}{4}}\end{array}$

Therefore, the required expressions are $x^{\frac{5}{4}}\text{ and }{x}^{\frac{1}{2}}$ .