Chapter 3.2, Problem 74E

Solution

The simplified form of the expression ${\left(x^5{y}^{-3}\right)}^{-0.25}$ .

  $\frac{y^{0.75}}{x^{1.5}}$

Given:

The expression ${\left(x^5{y}^{-3}\right)}^{-0.25}$ .

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 ${\left(x^5{y}^{-3}\right)}^{-0.25}$ , use the exponent properties and combine the like terms to simplify further as shown below,

  $\begin{array}{c}{\left(x^5{y}^{-3}\right)}^{-0.25}={\left(x^5\cdot \frac{1}{y^3}\right)}^{-0.25}\\ ={\left(\frac{x^5}{y^3}\right)}^{-0.25}\\ ={\left(\frac{y^3}{x^5}\right)}^{0.25}\\ =\frac{{\left(y^3\right)}^{0.25}}{{\left(x^5\right)}^{0.25}}\\ =\frac{y^{3\times 0.25}}{x^{5\times 0.25}}\\ =\frac{y^{0.75}}{x^{1.5}}\end{array}$

Therefore, the simplified expression is $\frac{y^{0.75}}{x^{1.5}}$ .