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Chapter 7: Problem 49

For exercises 39-82, simplify. $$ \frac{9 x^{2}}{8 y} \div \frac{x}{y} $$

Short Answer

Expert verified
\frac{9x}{8}

Step by step solution

01

- Rewrite the Division

Rewrite the division of fractions as a multiplication by the reciprocal. The division \(\frac{9 x^{2}}{8 y} \div \frac{x}{y}\) becomes \(\frac{9 x^{2}}{8 y} \times \frac{y}{x}\).
02

- Multiply the Numerators and Denominators

Multiply the numerators together and the denominators together: \(\frac{9 x^{2} \times y}{8 y \times x}\).
03

- Simplify the Expression

Combine the terms in both the numerator and the denominator: \(\frac{9 x^{2} y}{8 y x}\).
04

- Cancel Common Factors

Cancel out the common terms in the numerator and the denominator. The \(y\) term cancels out, leaving: \(\frac{9 x^{2}}{8 x} \). Then cancel one \(x\) term from the numerator and denominator resulting in \(\frac{9 x}{8}\).

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Key Concepts

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

Algebraic Manipulation
Algebraic manipulation involves rearranging and simplifying equations and expressions to make them easier to work with. One common tactic is to transform division into multiplication by using the reciprocal of the divisor. For example, when you have a problem like \(\frac{9 x^{2}}{8 y} \div \frac{x}{y}\), you can rewrite it as \(\frac{9 x^{2}}{8 y} \times \frac{y}{x}\). This makes it simpler to manage and sets up the problem for the next steps. Always rewrite division problems this way to pave the way for easier multiplication and further simplification.
Fraction Division
Fraction division is made easier by converting to multiplication. When you divide by a fraction, like \(\frac{9 x^{2}}{8 y} \div \frac{x}{y}\), it's equivalent to multiplying by its reciprocal. The reciprocal of \(\frac{x}{y}\) is \(\frac{y}{x}\). So the rewritten problem becomes \(\frac{9 x^{2}}{8 y} \times \frac{y}{x}\).
This step is crucial because multiplying fractions is much simpler and it aligns with familiar multiplication rules, allowing you to streamline the problem. In effect, changing division into multiplication simplifies the operation and sets you up for easier numerical or algebraic manipulation.
Simplifying Expressions
Simplifying expressions means breaking them down to their simplest form. After rewriting our problem from division to multiplication, you'll multiply the numerators together and the denominators together. So, \(\frac{9 x^{2}}{8 y} \times \frac{y}{x}\) becomes \(\frac{9 x^{2} \times y}{8 y \times x}\).
Next comes combining terms. Multiply the variables and constants in the numerator and do likewise for the denominator. This results in \(\frac{9 x^{2} y}{8 y x}\). Simplified expressions are easier to understand and work with, and this step ensures everything is in the simplest, most manageable form.
Canceling Common Factors
The final step in simplifying algebraic fractions is canceling out common factors. In the expression \(\frac{9 x^{2} y}{8 y x}\), notice the \( y \) in the numerator and denominator cancels out, leaving \(\frac{9 x^{2}}{8 x} \). Then, cancel one \( x \) from the numerator and denominator, leading to \(\frac{9 x}{8} \).
Canceling common factors reduces the expression to its simplest form. It's a critical step because it removes redundant terms, making the result cleaner and easier to understand. Always look for common factors between the numerator and denominator to simplify your expression to the fullest.

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Most popular questions from this chapter

The relationship of the number of tickets sold, \(x\), and the total ticket receipts for an outdoor concert, \(y\), is a direct variation. When 11,000 tickets are sold, the total ticket receipts are \(\$ 495,000\). a. Find the constant of proportionality, \(k\). Include the units of measurement. b. Write an equation that represents this relationship. c. Find the number of tickets sold when the total ticket receipts are \(\$ 562,500\). d. Find the total ticket receipts from the sale of 7575 tickets. e. What does \(k\) represent in this equation?

For exercises 61-64, the completed problem has one mistake. (a) Describe the mistake in words or copy down the whole problem and highlight or circle the mistake. (b) Do the problem correctly. Problem: The relationship of the number of weeks a box of garbage bags is used, \(x\), and the number of bags left in the box, \(y\), is an inverse variation. When \(x\) is 8 weeks, \(y\) is 168 bags. Find the constant of proportionality, \(k\). Incorrect Answer: \(k=\frac{y}{x}\) $$ k=\frac{168 \text { bags }}{8 \text { weeks }} $$

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)

For exercises 59-66, use the five steps. Assume that the rate of work does not change if done individually or together. The water from a garden hose turned on at full pressure fills a hot tub in \(45 \mathrm{~min}\). If the drain is open, the hot tub empties in \(62 \mathrm{~min}\). Find the amount of time to fill the hot tub with the drain open. Round to the nearest whole number.

For exercises \(45-48\), the formula \(R=\frac{U F}{P}\) describes the glomular filtration rate by a kidney \(R\). Is the relationship of the given variables a direct variation or an inverse variation? $$ U \text { and } F \text { are constant; the relationship of } R \text { and } P \text {. } $$
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