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Chapter 6: Problem 18

Simplify by removing a factor equal to 1. $$ \frac{35 n}{5 n^{2}} $$

Short Answer

Expert verified
\( \frac{7}{n} \)

Step by step solution

01

Factor numerator and denominator

Express the numerator and the denominator in terms of their prime factors. The numerator is given by: 35n. Factorizing 35 gives: 5 * 7. Therefore, the numerator becomes: 7 * 5 * n. The denominator is given by: 5n^2. This can be rewritten as: 5 * n * n.
02

Identify common factors

Notice that both the numerator and the denominator have the common factors 5n. These common factors can be simplified out.
03

Cancel out the common factors

Remove the common factors (5n) from both the numerator and the denominator. The expression then simplifies to: \( \frac{7}{n} \).
04

Write the simplified expression

The factorial reduction results in \( \frac{7}{n} \). This is the simplified form of the given expression.

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Key Concepts

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

Factoring
Factoring is the first step to simplifying rational expressions. It means breaking down a number or expression into its multiplicative components. For example, the number 35 can be factored into 5 and 7 because 35 = 5 * 7. In our exercise, we started by factoring both the numerator and the denominator.
Cancel Common Factors
Canceling common factors is a crucial step in simplifying rational expressions. After factoring both the numerator and the denominator, you look for any factors that are shared between them. In our case, the common factors were 5 and n. Once we identify these common factors, they can be canceled out, making our expression simpler. Remember, canceling out a factor is like dividing both the numerator and the denominator by that factor, which simplifies the overall fraction.
Prime Factors
Prime factors are the building blocks of numbers. A prime factor is a factor that is a prime number, which means it can only be divided by one and itself. When simplifying expressions, identifying prime factors can make the process easier. For instance, in the exercise provided, we broke down 35 into two prime factors: 5 and 7. Recognizing these prime factors helped us see the common factors between the numerator and the denominator, ultimately aiding in simplifying the expression.

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