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Chapter 7: Problem 23

Simplify by removing a factor equal to 1. $$ \frac{15 x}{5 x^{2}} $$

Short Answer

Expert verified
\( \frac{3}{x} \)

Step by step solution

01

- Understand the Problem

The given problem is \( \frac{15 x}{5 x^{2}} \). The task is to simplify this expression by removing a factor equal to 1.
02

- Factor the Numerator and Denominator

Factor both the numerator and the denominator: \(15x = 3 \times 5 \times x\) and \(5x^2 = 5 \times x \times x\).
03

- Simplify Common Factors

Cancel the common factors in the numerator and the denominator: \( \frac{3 \cdot 5 \cdot x}{5 \cdot x \cdot x} \). This simplifies to \( \frac{3}{x} \) after cancelling one 5 and one x.

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Key Concepts

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

Numerator and Denominator
In any rational expression, the top part is called the numerator, and the bottom part is called the denominator.
The numerator represents the number being divided, while the denominator represents the number you are dividing by.
For example, in the expression \(\frac{15x}{5x^2}\), 15x is the numerator and 5x^2 is the denominator.
Understanding the roles of the numerator and denominator is key to simplifying your expression.
Knowing this distinction helps you identify common factors and simplify the entire expression.
Common Factors
Common factors are numbers or variables that appear in both the numerator and the denominator.
To simplify a rational expression, you need to identify these common factors and cancel them out.
For instance, in the expression \(\frac{15x}{5x^2}\), both the numerator and the denominator include the factors of 5 and x.
By finding these common factors, you can break down and simplify the expression.
Always remember that canceling common factors is essential for simplifying rational expressions.
Factoring
Factoring is the process of breaking down an expression into simpler 'parts' or factors that can be multiplied to get the original expression.
This is crucial in simplifying rational expressions, as it helps to identify common factors.
For example, you can factor the numerator 15x as 3 * 5 * x and the denominator 5x^2 as 5 * x * x.
This allows you to see common factors easily and cancel them out.
After factoring, you can cancel the common factors from both the numerator and the denominator, simplifying your expression efficiently.

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Most popular questions from this chapter

The intensity I of a television signal varies inversely as the square of the distance \(d\) from the transmitter. If the intensity is \(25 \mathrm{W} / \mathrm{m}^{2}\) at a distance of \(2 \mathrm{km},\) what is the intensity \(6.25 \mathrm{km}\) from the transmitter?

Write an equivalent expression without parentheses. $$-(3-a)$$

The height of a triangle is \(3 \mathrm{cm}\) longer than its base. If the area of the triangle is \(54 \mathrm{cm}^{2},\) find the measurements of the base and the height.

Graph the function given by $$ f(x)=\frac{x^{2}-9}{x-3} $$ (Hint. Determine the domain of \(f\) and simplify.)

If \(y\) varies directly as \(x,\) does doubling \(x\) cause \(y\) to be doubled as well? Why or why not?
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