Chapter 9.9, Problem 11MRE

To determine

To simplify the expression ${\left(\frac{x}{y^2}\right)}^{-1}.{\left(\frac{y}{x^2}\right)}^2$

Answer to Problem 11MRE

  $\frac{y^4}{x^5}$ is the solution.

Explanation of Solution

Given: expression ${\left(\frac{x}{y^2}\right)}^{-1}.{\left(\frac{y}{x^2}\right)}^2$

Formula Used:

The rule of division for exponents with the same base states that when we divide the powers with the same base, we subtract the exponents.

Calculation:

Given: expression ${\left(\frac{x}{y^2}\right)}^{-1}.{\left(\frac{y}{x^2}\right)}^2$

Using the rule of BODMAS and the rule of multiplication of the exponents, the above expression can be simplied as follows:

  $\left(\frac{x^{-1}}{y^{-2}}\right).\left(\frac{y^2}{x^4}\right)$

Further, we use the rule of division for exponents with the same base; when we divide the powers, we subtract their exponents. Hence, the above expression can be further simplified into:

  $x^{\left(-1-4\right)}.y^{\left(2+2\right)}$

This further simplifies into

  $x^{-5}.y^4$

The above expression can be written as:

  $\frac{y^4}{x^5}$

This is the most simplified version.

Conclusion:

Hence, the solution is $\frac{y^4}{x^5}$ .