Chapter 3.2, Problem 72E

Solution

The simplified form of the expression ${\left(x^6{y}^2\right)}^{-0.75}$ .

  $\frac{1}{x^{4.5}{y}^{1.5}}$

Given:

The expression ${\left(x^6{y}^2\right)}^{-0.75}$ .

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 ${\left(x^6{y}^2\right)}^{-0.75}$ , use the exponent properties and combine the like terms to simplify further as shown below,

  $\begin{array}{c}{\left(x^6{y}^2\right)}^{-0.75}=\frac{1}{{\left(x^6{y}^2\right)}^{0.75}}\\ =\frac{1}{{\left(x^6\right)}^{0.75}\cdot {\left(y^2\right)}^{0.75}}\\ =\frac{1}{x^{6\times 0.75}\cdot y^{2\times 0.75}}\\ =\frac{1}{x^{4.5}\cdot y^{1.5}}\\ =\frac{1}{x^{4.5}{y}^{1.5}}\end{array}$

Therefore, the simplified expression is $\frac{1}{x^{4.5}{y}^{1.5}}$ .