Chapter 1.2, Problem 93E

(a)

To determine

To identify: the sign of expression $b^5$ .

Answer to Problem 93E

  $b^5<0$

Explanation of Solution

Given information:

Three real numbers $a,b,c$ such that $a>0,b<0$ , and $c<0$ .

Calculation:

Consider the given expression.

  $b^5$

Now, $b<0$

So, sign of $b^5$ can be determined as shown:

  $b^5=b\cdot b\cdot b\cdot b\cdot b$

If negative sign is multiplied odd times, the result becomes negative.

Here, in the expression $b^5$ , b is multiplied 5 times.

So, $b^5<0$ .

(b)

To determine

To identify: the sign of expression $b^{10}$ .

Answer to Problem 93E

  $b^{10}>0$

Explanation of Solution

Given information:

Three real numbers $a,b,c$ such that $a>0,b<0$ , and $c<0$ .

Calculation:

Consider the given expression.

  $b^{10}$

Now, $b<0$

So, sign of $b^{10}$ can be determined as shown:

  $b^5=b\cdot b\cdot b\cdot b\cdot b\cdot b\cdot b\cdot b\cdot b\cdot b$

If negative sign is multiplied even times, the result becomes negative.

Here, in the expression $b^{10}$ , b is multiplied 10 times.

So, $b^{10}>0$ .

(c)

To determine

To identify: the sign of expression $a{b}^2{c}^3$ .

Answer to Problem 93E

  $a{b}^2{c}^3<0$

Explanation of Solution

Given information:

Three real numbers $a,b,c$ such that, $a>0,b<0$ and $c<0$ .

Calculation:

Consider the given expression.

  $a{b}^2{c}^3$

Now, $a>0,b<0$ and $c<0$ .

So, sign of $a{b}^2{c}^3$ can be determined as shown:

  $\begin{array}{c}(+){(-)}^2{(-)}^3=(+)(+)(-)\\ =(+)(-)\\ =(-)\end{array}$

So, $a{b}^2{c}^3<0$ .

(d)

To determine

To identify: the sign of expression ${\left(b-a\right)}^3$ .

Answer to Problem 93E

  ${\left(b-a\right)}^3<0$

Explanation of Solution

Given information:

Three real numbers $a,b,c$ such that, $a>0,b<0$ and $c<0$ .

Calculation:

Now, $a>0,b<0$ and $c<0$ .

So, sign of ${\left(b-a\right)}^3$ can be determined as shown:

Consider the given expression.

  ${\left(b-a\right)}^3$

  $\begin{array}{c}{\left(--(+)\right)}^3={(-)}^3\\ =-\end{array}$

So, ${\left(b-a\right)}^3<0$ .

(e)

To determine

To identify: the sign of expression ${\left(b-a\right)}^4$ .

Answer to Problem 93E

  ${\left(b-a\right)}^4>0$

Explanation of Solution

Given information:

Three real numbers $a,b,c$ such that, $a>0,b<0$ and $c<0$ .

Calculation:

Consider the given expression.

  ${\left(b-a\right)}^4$

Now, $a>0,b<0$ and $c<0$ .

So, sign of ${\left(b-a\right)}^4$ can be determined as shown:

  $\begin{array}{c}{\left(--(+)\right)}^4={(-)}^4\\ =(+)\end{array}$

So, ${\left(b-a\right)}^4>0$ .

(f)

To determine

To identify: the sign of expression $\frac{a^3{c}^3}{b^6{c}^6}$ .

Answer to Problem 93E

  $\frac{a^3{c}^3}{b^6{c}^6}<0$

Explanation of Solution

Given information:

Three real numbers $a,b,c$ such that, $a>0,b<0$ and $c<0$ .

Calculation:

Consider the given expression.

  $\frac{a^3{c}^3}{b^6{c}^6}$

Now, $a>0,b<0$ and $c<0$ .

So, sign of $\frac{a^3{c}^3}{b^6{c}^6}$ can be determined as shown:

  $\begin{array}{c}\frac{{(+)}^3{(-)}^3}{{(-)}^6{(-)}^6}=\frac{(+)(-)}{(+)(+)}\\ =\frac{(-)}{(+)}\\ =(-)\end{array}$

So, $\frac{a^3{c}^3}{b^6{c}^6}<0$ .