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Chapter 7: Problem 11

For exercises 7-32, simplify. $$ \left(\frac{x^{2}+5 x}{x^{2}}\right)\left(\frac{3 x}{x+5}\right) $$

Short Answer

Expert verified
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Step by step solution

01

- Rewrite the Expression

Write the given expression as a single fraction by multiplying the numerators and the denominators: \[ \frac{(x^2 + 5x) \times 3x}{x^2 \times (x + 5)} \]
02

- Factor Common Terms

Factor common terms from the numerator. Notice that the numerator \(x^2 + 5x\) can be written as \(x(x + 5)\): \[ \frac{x(x + 5) \times 3x}{x^2 \times (x + 5)} \]
03

- Simplify the Fraction

Cancel the common factors in the numerator and the denominator. The \(x + 5\) in the numerator and denominator cancel each other out: \[ \frac{x \times 3x}{x^2} = \frac{3x^2}{x^2} \]
04

- Final Simplification

Simplify \( \frac{3x^2}{x^2} \): \[ 3 \]

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Key Concepts

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

Factoring
Factoring is a method used to break down complex algebraic expressions into simpler, multiplyable components.
This is essential when simplifying expressions or solving equations.
For example, consider the expression \(x^2 + 5x\).
Here, you can factor it by identifying a common term, which is \(x\), and rewrite it as \(x(x + 5)\).
Factoring often simplifies expressions by making it easier to identify and subsequently cancel common terms.
Canceling Common Terms
Canceling common terms involves reducing a fraction by dividing both the numerator and the denominator by the same factor.
This helps in simplifying algebraic expressions effectively.
The key is to first factorize both the numerator and the denominator.
In our example, after factoring we have: \[ \frac{x(x + 5) \times 3x}{x^2 \times (x + 5)} \]
Here, \((x + 5)\) is a common term that appears both in the numerator and the denominator. Hence, it can be canceled out, leaving us with: \[ \frac{x \times 3x}{x^2} \]
Notice how canceling common terms significantly simplifies the expression.
Multiplying Fractions
Multiplying fractions involves multiplying the numerators together and the denominators together.
This is a fundamental skill in simplifying algebraic expressions that involve ratios. Consider the initial expression given: \[ \frac{x^{2}+5 x}{x^{2}} \times \frac{3 x}{x+5} \]
The process starts by multiplying the numerators \( (x^2 + 5x) \times 3x \) and the denominators \( x^2 \times (x + 5) \)
Written as one combined fraction: \[ \frac{(x^2 + 5x) \times 3x}{x^2 \times (x + 5)} \]
Once the product of numerators and denominators are written, the steps of factoring and canceling common terms follow to simplify the expression.
Algebraic Simplification
Algebraic simplification is the process of making an algebraic expression as simple as possible.
This involves steps like factoring, canceling common terms, and sometimes even rearranging the terms.
In our case, after canceling common terms, we were left with: \[ \frac{3x^2}{x^2} \]
Here, \(x^2\) in the numerator and denominator cancel each other out, leaving us with just the coefficient, \(3\).
The final simplified expression is then: \[3 \]
Simplification makes it easier to understand and solve algebraic problems by reducing them to their simplest form.

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Most popular questions from this chapter

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)

If the annual credit sales are constant, the relationship of the accounts receivable, \(x\), and the financial ratio receivables turnover, \(y\), is an inverse variation. The accounts receivable of a company are \(\$ 150\) million, and its receivables turnover ratio is 12 . a. Find the constant of proportionality. Include the units of measurement. b. Write an equation that represents this relationship. c. Find the receivables turnover ratio when the accounts receivable are \(\$ 200\) million.

The relationship of the taxable value of a property, \(x\), and the annual property tax, \(y\), is a direct variation. When the taxable value of a property is \(\$ 250,000\), the annual property tax bill is \(\$ 5375\). a. Find the constant of proportionality, \(k\). b. Write an equation that represents this relationship. c. Find the taxable value of a property with an annual property tax bill of \(\$ 8062.50\). d. Find the tax owed for a property with an assessed value of \(\$ 185,000\). Round to the nearest whole number. e. What does \(k\) represent in this equation?

For exercises 43-58, (a) solve. (b) check. $$ \frac{9}{10} v+\frac{1}{3}=-\frac{22}{15} $$

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.
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