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Chapter 7: Problem 18

For exercises \(9-24\), evaluate or simplify. $$ \frac{\frac{r^{2}+11 r+24}{9 r}}{\frac{r^{2}-64}{27 r^{3}}} $$

Short Answer

Expert verified
The simplified expression is \( \frac{3r^{2}(r+3)}{(r-8)} \).

Step by step solution

01

- Rewrite the Complete Expression

Start by rewriting the given complex fraction as a division problem: \[ \frac{\frac{r^{2}+11r+24}{9r}}{\frac{r^{2}-64}{27r^{3}}} = \frac{\frac{r^{2}+11r+24}{9r}}{\frac{r^{2}-64}{27r^{3}}} = \frac{r^{2}+11r+24}{9r} \times \frac{27r^{3}}{r^{2}-64}. \]
02

- Factor the Polynomials

Factor the quadratic expressions in the numerator and the denominator. \[ r^{2}+11r+24 = (r+3)(r+8), \ r^{2}-64 = (r-8)(r+8). \] Substituting back, we get: \[ \frac{(r+3)(r+8)}{9r} \times \frac{27r^{3}}{(r-8)(r+8)}. \]
03

- Cancel Common Terms

Look for common terms that can be cancelled in the numerator and denominator: \[ \frac{(r+3)(r+8)}{9r} \times \frac{27r^{3}}{(r-8)(r+8)} = \frac{(r+3)}{9} \times \frac{27r^{2}}{(r-8)} = \frac{(r+3) \times 27r^{2}}{9 \times (r-8)}. \]
04

- Simplify the Remaining Expression

Perform the necessary multiplication and division: \[ \frac{(r+3) \times 27r^{2}}{9 \times (r-8)} = 3r^{2} \times \frac{(r+3)}{(r-8)} = \frac{27r^{2}(r+3)}{9(r-8)} = \frac{3r^{2}(r+3)}{(r-8)}. \]

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Key Concepts

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

Factoring Polynomials
Factoring polynomials is like breaking down a complex structure into simpler building blocks. When you factor a polynomial, you express it as a product of its simpler factors.

For example, consider the quadratic polynomial \(r^{2} + 11r + 24\). It can be factored into two binomials: \((r + 3)(r + 8)\). Similarly, a difference of squares like \(r^{2} - 64\) can be factored into \((r - 8)(r + 8)\). Recognizing these patterns helps simplify expressions and solve equations more efficiently.
Multiplying Fractions
When multiplying fractions, you multiply the numerators together and the denominators together.

For the complex fraction \(\frac{\frac{r^{2}+11r+24}{9r}}{\frac{r^{2}-64}{27r^{3}}}\), we first rewrite it as a division problem, which involves multiplying by the reciprocal.

This turns into: \(\frac{r^{2}+11r+24}{9r} \times \frac{27r^{3}}{r^{2}-64}\). Now, multiply the individual numerators and denominators to get the complete multiplied fraction.
Canceling Common Factors
One of the most useful simplifications in algebra involves canceling common factors. After factoring the numerators and denominators, look for terms that are exactly the same and appear both in the numerator and the denominator.

In our example, factor the terms: \(\frac{(r+3)(r+8)}{9r} \times \frac{27r^{3}}{(r-8)(r+8)}\). The \((r+8)\) terms in the numerator and denominator cancel out, leaving you with: \(\frac{(r+3) \times 27r^{2}}{9 \times (r-8)}\). Recognizing and canceling these common factors simplifies the problem considerably.
Algebraic Expressions
Algebraic expressions are combinations of variables, numbers, and operation symbols. Simplifying them often involves many steps, including factoring, multiplying, and canceling terms.

Consider \(\frac{(r+3) \times 27r^{2}}{9 \times (r-8)}\). Simplify each part by performing the operations: \(27r^{2}\) divided by \(9\) simplifies to \(3r^{2}\). So, the expression becomes: \(3r^{2} \times \frac{(r+3)}{(r-8)}\). The final simplified form is \(\frac{3r^{2}(r+3)}{(r-8)}\). Understanding these steps helps in dealing with any algebraic expression efficiently.

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Most popular questions from this chapter

In 2011, the total property tax millage rate for Fort Lauderdale, Florida, was \(20.1705\). (For every \(\$ 1000\) in taxable property, an owner owes a tax of \(\$ 20.1705\).) If a property owner pays the tax in four installments, a discount is applied to the first three installments. Find the total amount of tax paid by installments on taxable property of \(\$ 175,000\). Round to the nearest hundredth. $$ \begin{array}{|c|c|} \hline \text { Installment due date } & \text { Discount on the payment } \\ \hline \text { June 30 } & 6 \% \\ \text { September 30 } & 4.5 \% \\ \text { December 31 } & 3 \% \\ \text { March 31 } & \text { None } \\ \hline \end{array} $$ Sources: www.broward.org; www.bcpa.net.millage.asp

For exercises \(67-82\), use the five steps and a proportion. Find the number of 725,000 women in their mid \(-40 \mathrm{~s}\) with a history of normal pregnancy who would be expected to have a heart attack or stroke some 10 years later. Of 100 women in their mid-40's with a history of normal pregnancy, about 4 would be expected to have a heart attack or stroke some 10 years later. (Source: www.nytimes.com, March 17, 2009)

The relationship of the radius of a circle, \(x\), and the circumference of the circle, \(y\), is a direct variation. The radius of a circle is \(10 \mathrm{~cm}\), and the circumference is \(62.8 \mathrm{~cm}\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this relationship. c. Find the circumference of a circle with a radius of \(20 \mathrm{~cm}\).

Explain why the relationship of the number of bags of leaves per hour that are raked, \(x\), and the hours it takes to rake a yard, \(y\), is an inverse variation.

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)
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