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Chapter 4: Problem 10

For Problems \(9-50\), simplify each rational expression. $$ \frac{21 x y}{35 x} $$

Short Answer

Expert verified
The simplified expression is \(\frac{3y}{5}\).

Step by step solution

01

Factor the Numerator and Denominator

First, factor both the numerator and the denominator of the fraction. The numerator is \(21xy\), which factors into \(3 \times 7 \times x \times y\). The denominator is \(35x\), which factors into \(5 \times 7 \times x\).
02

Cancel Common Factors

Look for common factors in the numerator and the denominator. Here, both the numerator and the denominator have the factors \(7\) and \(x\). Cancel out these common factors. After canceling, we are left with \(\frac{3y}{5}\).
03

Write the Simplified Fraction

The fraction is now fully simplified. After canceling the common factors, we have the simplified expression \(\frac{3y}{5}\).

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Key Concepts

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

Factoring Algebraic Expressions
Before simplifying a rational expression, it's essential to understand how to factor both the numerator and the denominator fully. This means breaking down the algebraic components into their simplest form. In our example, the expression was \(\frac{21xy}{35x}\). To factor this:
  • The numerator \(21xy\) was factored into \(3 \times 7 \times x \times y\).
  • The denominator \(35x\) was factored into \(5 \times 7 \times x\).
Factoring allows us to see the individual terms clearly and helps us identify any common factors between the numerator and the denominator. This is a critical step before reducing any expression. By getting comfortable with this process, future simplifications will become easier and more intuitive.
Canceling Common Factors
After factoring both parts of the rational expression, the next step is to cancel out any common factors. This is essentially removing any terms that appear in both the numerator and the denominator. This step simplifies the fraction and brings it closer to its simplest form. For the example, \(\frac{3 \times 7 \times x \times y}{5 \times 7 \times x}\), the common factors are:
  • 7 in both the numerator and the denominator
  • x in both the numerator and the denominator
Canceling these out leaves just \(\frac{3y}{5}\). It's like cleaning up the expression to remove unnecessary clutter and it's important to only do this where the factors are exactly the same. This process is essential in maintaining the equality of the fraction while making it more straightforward.
Simplified Expression
Once common factors have been canceled, what remains is the simplified expression. The process of simplification transforms the original fraction into its most reduced and clear form. After canceling the common factors from the provided example, the expression simplifies to \(\frac{3y}{5}\).
At this point:
  • No further factors can be canceled since 3, y, and 5 share no common factors.
  • The expression \(\frac{3y}{5}\) is as simple as possible and is equivalent to the original expression \(\frac{21xy}{35x}\).
The goal in simplifying is to make the expression easier to work with while maintaining its original value. The benefits are improved clarity and efficiency, especially when dealing with more complex equations in future calculations.

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Most popular questions from this chapter

For Problems \(1-30\), solve each equation. $$ \frac{x}{3 x-6}+\frac{4}{x^{2}-4}=\frac{1}{3} $$

Felipe jogs for 10 miles and then walks another 10 miles. He jogs \(2 \frac{1}{2}\) miles per hour faster than he walks, and the entire distance of 20 miles takes 6 hours. Find the rate at which he walks and the rate at which he jogs.

For Problems \(1-44\), solve each equation. $$ \frac{n+6}{27}=\frac{1}{n} $$

For Problems \(1-44\), solve each equation. $$ \frac{n}{n+1}+\frac{1}{2}=\frac{-2}{n+2} $$

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