Chapter 11.1, Problem 1P

(a)

To determine

Simplify the expression and state any excluded values.

Answer to Problem 1P

  $\frac{21a^2}{7a^3}=\frac{3}{a}$

Explanation of Solution

Given: $\text{a}:\frac{21a^2}{7a^3}$

Concept Used:

A rational expression is considered simplified if the numerator and denominator have no factors in common.

Step 1: Factor the numerator and the denominator.

Step 2: List restricted values.

Step 3: Cancel common factors.

Step 4: Simplify and note any restricted values not implied by the expression.

Rational expression is in the form $\frac{\text{P}}{\text{Q}}$ , where $\text{Q}\ne 0$ , denominator cannot be zero in any rational expression.

Excluded values are also called points of discontinuity. These are the values that make the denominator equal to zero and are not part of the domain.

In any rational function, the denominator cannot be zero.

To find the excluded value of the function, make denominator equal to zero and simplify.

  $\text{Quotient}\text{rule}:\frac{a^{ m}}{a^n}=a^{m-n}$

Calculation:

  $\text{a}:\frac{21a^2}{7a^3}=\frac{7·3a^2}{7a^3}=\frac{3}{a}[\frac{a^{ m}}{a^n}=a^{m-n}]$ Excluded value $\text{a}\ne 0$

Thus, $\frac{21a^2}{7a^3}=\frac{3}{a}$

(b)

To determine

Simplify the expression and state any excluded values.

Answer to Problem 1P

  $\frac{18d^2}{4d+8}=\frac{9d^2}{2(d+2)}$

Explanation of Solution

Given: $\text{b:}\frac{18d^2}{4d+8}$

Concept Used:

A rational expression is considered simplified if the numerator and denominator have no factors in common.

Step 1: Factor the numerator and the denominator.

Step 2: List restricted values.

Step 3: Cancel common factors.

Step 4: Simplify and note any restricted values not implied by the expression.

Rational expression is in the form $\frac{\text{P}}{\text{Q}}$ , where $\text{Q}\ne 0$ , denominator cannot be zero in any rational expression.

Excluded values are also called points of discontinuity. These are the values that make the denominator equal to zero and are not part of the domain.

In any rational function, the denominator cannot be zero.

To find the excluded value of the function, make denominator equal to zero and simplify.

Calculation:

  $\text{b:}\frac{18d^2}{4d+8}=\frac{18d^2}{4(d+2)}=\frac{9d^2}{2(d+2)}$ Excluded value $d+2\ne 0\Rightarrow d\ne -2$

Thus, $\frac{18d^2}{4d+8}=\frac{9d^2}{2(d+2)}$

(c)

To determine

Simplify the expression and state any excluded values.

Answer to Problem 1P

  $\frac{2n-3}{6n-9}=\frac{1}{3}$

Explanation of Solution

Given: $\text{c:}\frac{2n-3}{6n-9}$

Concept Used:

A rational expression is considered simplified if the numerator and denominator have no factors in common.

Step 1: Factor the numerator and the denominator.

Step 2: List restricted values.

Step 3: Cancel common factors.

Step 4: Simplify and note any restricted values not implied by the expression.

Rational expression is in the form $\frac{\text{P}}{\text{Q}}$ , where $\text{Q}\ne 0$ , denominator cannot be zero in any rational expression.

Excluded values are also called points of discontinuity. These are the values that make the denominator equal to zero and are not part of the domain.

In any rational function, the denominator cannot be zero.

To find the excluded value of the function, make denominator equal to zero and simplify.

Calculation:

  $\text{c:}\frac{2n-3}{6n-9}=\frac{2n-3}{3(2n-3)}=\frac{1}{3}$ ; No excluded value.

Thus, $\frac{2n-3}{6n-9}=\frac{1}{3}$

(d)

To determine

Simplify the expression and state any excluded values.

Answer to Problem 1P

  $\frac{26c^3+91c}{2c^2+7}=13c$

Explanation of Solution

Given: $\text{d:}\frac{26c^3+91c}{2c^2+7}$

Concept Used:

A rational expression is considered simplified if the numerator and denominator have no factors in common.

Step 1: Factor the numerator and the denominator.

Step 2: List restricted values.

Step 3: Cancel common factors.

Step 4: Simplify and note any restricted values not implied by the expression.

Rational expression is in the form $\frac{\text{P}}{\text{Q}}$ , where $\text{Q}\ne 0$ , denominator cannot be zero in any rational expression.

Excluded values are also called points of discontinuity. These are the values that make the denominator equal to zero and are not part of the domain.

In any rational function, the denominator cannot be zero.

To find the excluded value of the function, make denominator equal to zero and simplify.

Calculation:

  $\text{d:}\frac{26c^3+91c}{2c^2+7}=\frac{13c(2c^2+7)}{2c^2+7}=13c$ ; No excluded value.

Thus, $\frac{26c^3+91c}{2c^2+7}=13c$