Chapter 14, Problem 27CCSR

Solution

a.

To simplify: the given expression ${\left(\frac{r^{3m}}{r^{-m}{t}^{4n}}\right)}^{\frac{1}{n}}\cdot {\left(\frac{r^{\frac{1}{n}}}{t^{\frac{2}{m}}}\right)}^{-m}$ .

The simplified expression is $\left(\frac{r^{\frac{3m}{n}}}{t^2}\right)$ .

Given information:

The given expression is ${\left(\frac{r^{3m}}{r^{-m}{t}^{4n}}\right)}^{\frac{1}{n}}\cdot {\left(\frac{r^{\frac{1}{n}}}{t^{\frac{2}{m}}}\right)}^{-m}$ .

Concept Used:

Calculation:

Use the power property of exponent: ${\left(a^m\right)}^n=a^{mn}$ .

  $\left(\frac{r^{3m\cdot \frac{1}{n}}}{r^{-m\cdot \frac{1}{n}}{t}^{4n\cdot \frac{1}{n}}}\right)\cdot \left(\frac{r^{\frac{1}{n}\cdot -m}}{t^{\frac{2}{m}\cdot -m}}\right)=\left(\frac{r^{\frac{3m}{n}}}{r^{-\frac{m}{n}}{t}^4}\right)\left(\frac{r^{-\frac{m}{n}}}{t^{-2}}\right)$

Use the quotient property of exponent: $a^{m-n}=\frac{a^m}{a^n}$ .

  $\left(\frac{r^{\frac{3m}{n}+\frac{m}{n}}}{t^4}\right)\left(\frac{r^{-\frac{m}{n}}}{t^{-2}}\right)=\left(\frac{r^{\frac{4m}{n}}}{t^4}\right)\left(\frac{r^{-\frac{m}{n}}}{t^{-2}}\right)$

Use the product property of exponent: $a^{m+n}=a^m\cdot a^n$

  $\left(\frac{r^{\frac{4m}{n}-\frac{m}{n}}}{t^{4-2}}\right)=\left(\frac{r^{\frac{3m}{n}}}{t^2}\right)$

b.

To determine: the value of the expression found in part (a) when $m=1,n=2$ and $t=-3i$ .

The value of the expression found in part(a) when $m=1,n=2$ and $t=-3i$ is $-\frac{r^{\frac{3}{2}}}{9}$ .

Given information:

The simplified expression is $\left(\frac{r^{\frac{3m}{n}}}{t^2}\right)$ .

Concept Used:

Substitution means putting numbers in place of letters to calculate the value of an expression.

Calculation:

Substitute $1$ for $m$ , $2$ for $n$ and $-3i$ for $t$ in $\left(\frac{r^{\frac{3m}{n}}}{t^2}\right)$ .

  $\left(\frac{r^{\frac{3(1)}{(2)}}}{{(-3i)}^2}\right)=-\frac{r^{\frac{3}{2}}}{9}$

c.

To explain: the value of $r$ will the expression you found in part(b) be a real number.

The expression is a real number when the value of $r$ is any non-negative real number.

Given information:

The value of the expression found in part(a) when $m=1,n=2$ and $t=-3i$ is $-\frac{r^{\frac{3}{2}}}{9}$ .

Concept Used:

A non-negative number is a number which is either zero or a positive number 

Calculation:

The expression $\sqrt{r}$ is a real number when the value of $r$ is a non-negative real number.

The expression $r$ is a real number when the value of $r$ is a non-negative real number.