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Chapter 4: Problem 590

In the following exercises, simplify. $$-\frac{14 a}{14 b}$$

Short Answer

Expert verified
-\frac{a}{b}

Step by step solution

01

Factor out constants

Both the numerator and the denominator have a common factor of 14. Factor out the 14 from both the numerator and the denominator.
02

Simplify the fraction

After factoring out the common factor of 14, the expression inside the fraction becomes \[-\frac{14 \times a}{14 \times b}\]. We can cancel the 14 in the numerator and the denominator.
03

Divide by the common factor

Dividing the common factor of 14 out results in \[-\frac{a}{b}\].

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Key Concepts

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

common factors
In mathematics, a common factor is a number that divides two or more numbers without leaving a remainder. When simplifying fractions, finding common factors is key. For example, in the fraction \(\frac{14a}{14b}\), both the numerator (top part) and the denominator (bottom part) share the common factor 14.

When you recognize these common factors, you can divide both parts of the fraction by this number to simplify it.

Always look for the greatest common factor (GCF) to simplify the fraction as much as possible. In this instance, the GCF is 14, and simplifying gives you \(\frac{a}{b}\). Remember, simplifying makes calculations easier and results cleaner.
numerator and denominator
Fractions are composed of two main parts: the numerator and the denominator.

The numerator is the number on top of the fraction. It represents how many parts of the whole you are considering.

The denominator is the number at the bottom. It shows into how many equal parts the whole is divided.

Understanding these parts helps in simplifying and manipulating fractions.
In our example, \(\frac{14a}{14b}\), 14a is the numerator and 14b is the denominator.

By focusing on their relationship and common factors,
you simplify fractions efficiently.
dividing fractions
Dividing fractions involves inverting the second fraction (taking its reciprocal) and then multiplying the result. However, when simplifying fractions, we often divide the numerator and the denominator by a common factor.

For example, with \(\frac{14a}{14b}\), you recognize that both terms share 14.
By dividing both by 14, we are effectively simplifying the fraction.

Simplifying fractions using common factors maintains the same ratio but makes the numbers easier to work with. This process transforms \(\frac{14a}{14b}\) into \(-\frac{a}{b}\).
Remember, finding the greatest common factor (GCF) makes this process simpler and ensures your fractions are fully simplified.

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Most popular questions from this chapter

In the following exercises, subtract. $$\frac{11}{15}-\frac{5}{15}-\left(-\frac{2}{15}\right)$$

In the following exercises, solve the equation. $$ -z=23 $$

In the following exercises, simplify. $$ -\frac{3}{4} \div \frac{x}{3} $$

In the following exercises, subtract. $$\frac{3}{2}-\left(\frac{3}{2}\right)$$

In the following exercises, perform the indicated operation. $$ 5 \frac{8}{11}+2 \frac{4}{11} $$
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