Chapter 6, Problem 4MCQ

To determine

To Evaluate: The simplified form of the expression ${\left(\frac{p^2{r}^3}{p{r}^4}\right)}^2$ where $p\ne 0,r\ne 0$ .

Answer to Problem 4MCQ

The value of the expression ${\left(\frac{p^2{r}^3}{p{r}^4}\right)}^2$ is $\frac{p^2}{r^2}$.

Explanation of Solution

Given information:

The expression is ${\left(\frac{p^2{r}^3}{p{r}^4}\right)}^2$ .

Formula used:

For any real value $x$ and integer $a$ , $b$ , the power of power property states as ${\left(x^a\right)}^b=x^{ab}$

For any real value $x$ and integer $a$ , $b$ , if $x\ne 0$ the quotient of power property states as $\frac{x^b}{x^a}=x^{b-a}$

For any real value $x$ and integer $a$ , the negative exponent of $x$ that is $x^{-a}$ rewrite as $\frac{1}{x^a}$ .

Calculation:

Consider the expression is ${\left(\frac{p^2{r}^3}{p{r}^4}\right)}^2$ .

Recall that for any real value $x$ and integer $a$ , $b$ , the power of power property states as ${\left(x^a\right)}^b=x^{ab}$

Apply it,

  $\begin{array}{c}{\left(\frac{p^2{r}^3}{p{r}^4}\right)}^2=\frac{p^{2\times 2}{r}^{3\times 2}}{p^{1\times 2}{r}^{4\times 2}}\\ =\frac{p^4{r}^6}{p^2{r}^8}\end{array}$

Again recall that for any real value $x$ and integer $a$ , $b$ , if $x\ne 0$ the quotient of power property states as $\frac{x^b}{x^a}=x^{b-a}$

Use it,

  $\begin{array}{c}\frac{p^4{r}^6}{p^2{r}^8}=p^{4-2}{r}^{6-8}\\ =p^2{r}^{-2}\end{array}$

Again recall that for any real value $x$ and integer $a$ , the negative exponent of $x$ that is $x^{-a}$ rewrite as $\frac{1}{x^a}$ .

Further simplify,

  $p^2{r}^{-2}=\frac{p^2}{r^2}$

Thus, the value of the expression ${\left(\frac{p^2{r}^3}{p{r}^4}\right)}^2$ is $\frac{p^2}{r^2}$ .