Chapter P.1, Problem 51E

Solution

To simplify the expression $\frac{{\left(x^{-3}{y}^2\right)}^{-4}}{{\left(y^6{x}^{-4}\right)}^{-2}}$ .

  $x^4{y}^4$

Given:

The expression, $\frac{{\left(x^{-3}{y}^2\right)}^{-4}}{{\left(y^6{x}^{-4}\right)}^{-2}}$ .

Concept Used:

Exponent Properties:

• Product property: $a^m\cdot a^n=a^{m+n}$
• Quotient property: $\frac{a^m}{a^n}=a^{m-n}$
• Power of a power property: ${\left(a^m\right)}^n=a^{m\cdot n}$
• Power to product property: ${\left(ab\right)}^n=a^n\cdot b^n$
• Negative exponent property: $a^{-n}=\frac{1}{a^n}$

Calculation:

In order to solve the given expression $\frac{{\left(x^{-3}{y}^2\right)}^{-4}}{{\left(y^6{x}^{-4}\right)}^{-2}}$ , use the exponent properties and simplify further as shown below,

  $\begin{array}{c}\frac{{\left(x^{-3}{y}^2\right)}^{-4}}{{\left(y^6{x}^{-4}\right)}^{-2}}=\frac{{\left(\frac{1}{x^3}\cdot y^2\right)}^{-4}}{{\left(y^6\cdot \frac{1}{x^4}\right)}^{-2}}\\ =\frac{{\left(\frac{y^2}{x^3}\right)}^{-4}}{{\left(\frac{y^6}{x^4}\right)}^{-2}}\\ =\frac{{\left(\frac{x^3}{y^2}\right)}^4}{{\left(\frac{x^4}{y^6}\right)}^2}\\ =\frac{\frac{x^{3\cdot 4}}{y^{2\cdot 4}}}{\frac{x^{4\cdot 2}}{y^{6\cdot 2}}}\\ =\frac{\frac{x^{12}}{y^8}}{\frac{x^8}{y^{12}}}\\ =\frac{x^{12}}{y^8}\cdot \frac{y^{12}}{x^8}\\ =x^{12-8}\cdot y^{12-8}\\ =x^4{y}^4\end{array}$

Therefore, the simplified expression is $x^4{y}^4$ .