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Chapter 7: Problem 14

For exercises 7-32, simplify. $$ \left(\frac{r^{2}}{r^{2}+2 r}\right)\left(\frac{r^{2}-4}{r}\right) $$

Short Answer

Expert verified
\(\frac{r-2}{r}\)

Step by step solution

01

Factor the Expressions

Factor the expression \(\frac{r^{2}}{r^{2}+2r}\times\frac{r^{2}-4}{r}\). Start by factoring the denominator in \( \frac{r^{2}}{r^{2}+2r} \) and the numerator in \( \frac{r^{2}-4}{r} \). \( r^{2}+2r \) can be factored to \( r(r+2) \). \( r^{2}-4 \) is a difference of squares and can be factored to \( (r-2)(r+2) \). This results in: \(\frac{r^{2}}{r(r+2)} \) and \( \frac{(r-2)(r+2)}{r} \).
02

Multiply the Fractions

Combine the fractions by multiplying the numerators together and the denominators together: \[ \frac{r^{2}}{r(r+2)} \times \frac{(r-2)(r+2)}{r} = \frac{r^{2} \times (r-2)(r+2)}{r(r+2) \times r} \]
03

Simplify the Expression

Now simplify the expression by canceling out the common factors. \( r^{2} \) in the numerator and \( r \) in the denominator cancel out one \( r \). \( (r+2) \) in both the numerator and the denominator will cancel out. This leaves: \[ \frac{r(r-2)}{r^{2}} \]
04

Final Simplification

Finally, simplify \ \frac{r(r-2)}{r^{2}} \ by dividing both the numerator and the denominator by \( r \): \[ \frac{r(r-2)}{r^{2}} = \frac{r-2}{r} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Expressions
To simplify an algebraic fraction, a key step is factoring expressions. Factoring is the process of breaking down an expression into its simplest components. Consider the original fraction \(\frac{r^{2}}{r^{2}+2r}\times\frac{r^{2}-4}{r}\). To factor \(r^{2}+2r\), look for common factors. Here, \(r\) is common to both terms, so we factor it out: \(r^{2}+2r = r(r+2)\). Similarly, \(r^{2}-4\) is a special type of polynomial called a 'difference of squares'. It can be factored as \(r^{2}-4 = (r-2)(r+2)\). This step makes complex fractions easier to work with and sets us up for further simplification.
Multiplying Fractions
Once all parts of each fraction are factored, the next step is multiplying fractions. Multiplying fractions involves multiplying the numerators together and the denominators together. For our given problem, we'll multiply: \(\frac{r^{2}}{r(r+2)} \times \frac{(r-2)(r+2)}{r} = \frac{r^{2} \times (r-2)(r+2)}{r(r+2) \times r}\). Always check your factored form before multiplication. Correct setup prevents simple calculation errors and makes cancellation straightforward later on.
Cancelling Common Factors
After multiplying fractions, the next step is to simplify by canceling common factors. Canceling reduces fractions to their simplest forms and involves crossing out identical factors in the numerator and denominator. In our exercise, after multiplication, the result is \(\frac{r^{2}(r-2)(r+2)}{r(r+2)r}\). You can cancel \(r\) from \(r^{2}\) and \(r\) in the denominator, leaving \(r(r-2)(r+2)/(r(r+2))=r(r-2)/r\). Then, the \((r+2)\) terms cancel out: \(\frac{r(r-2)}{r^{2}} \rightarrow \frac{r-2}{r}\). Simplifying by canceling common factors turns complex expressions into manageable fractions.
Difference of Squares
In algebra, a 'difference of squares' is a special factoring case. It's crucial to recognize and use, especially with complex fractions. A difference of squares is any expression in the form \(a^{2} - b^{2}\), which factors into \((a-b)(a+b)\). For the given exercise, \(r^{2} - 4 = (r)^2 - (2)^2\). Following the difference of squares formula, it simplifies to \((r-2)(r+2)\). These techniques simplify expressions quickly and efficiently and are widely applicable in algebraic manipulations.

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Most popular questions from this chapter

When the top of a cone is removed, the formula for the volume of the remaining cone (the frustrum) is \(V=\frac{1}{3} \pi\left(R^{2}+R r+r^{2}\right) h\), where \(r\) is the radius of the circle at the top of the frustrum and \(R\) is the radius of the circle at the bottom of the frustrum. In 1856, an American army officer, Henry Hopkins Sibley, invented and received a patent for the design of a conical tent that could sleep 12 soldiers. (The apex is the diameter of the top of the frustrum.) Find the volume of the tent in cubic feet. Use \(\pi \approx 3.14\). Round to the nearest whole number. Be it known that I, H.H. Sibley, United States Army, have invented a new and improved Conical Tent ... the tent is in shape the frustrum of a cone; the base 18 feet; the height 12 feet; the apex 1 foot 6 inches [1.5 ft]. (Source: patimg1.uspto.gov)

The relationship of the amount of weed killer concentrate, \(x\), and the amount of mixed weed killer spray, \(y\), is a direct variation. A gardener uses \(2 \mathrm{oz}\) of concentrate to make 1 gal of weed killer spray. a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the amount of mixed weed killer spray that can be made with \(8 \mathrm{oz}\) of concentrate. d. Use slope-intercept graphing to graph this equation. e. Use the graph to find the amount of mixed weed killer spray that can be made with \(6 \mathrm{oz}\) of concentrate.

The relationship of \(x\) and \(y\) is a direct variation. When \(x=2, y=6\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this direct variation. c. Find \(y\) when \(x=4\). d. Use slope-intercept graphing to graph this equation. e. Use the graph to find \(y\) when \(x=5\).

A regulation basketball court in the NBA and the NCAA is \(94 \mathrm{ft}\) long and \(50 \mathrm{ft}\) wide. A regulation high school basketball court is \(84 \mathrm{ft}\) long and \(50 \mathrm{ft}\) wide. Find the percent increase in the area of an NCAA court compared to a high school court. Round to the nearest percent.

For exercises 11-30, (a) solve. (b) check. $$ \frac{3}{10}+\frac{7}{m}=\frac{14}{m}+\frac{1}{15} $$
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