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Chapter 2: Problem 66

For Problems \(65-74\), reduce each fraction to simplest form. $$ \frac{8 x^{2} y}{14 x} $$

Short Answer

Expert verified
\(\frac{8x^2y}{14x} = \frac{4xy}{7}\) in simplest form.

Step by step solution

01

Identify Common Factors

First, we need to find the common factors in both the numerator and the denominator of the fraction \( \frac{8x^2y}{14x} \). We see that both 8 and 14 have a common factor of 2, and both terms have at least one \(x\) in common.
02

Factor Out the Common Terms

Next, we'll factor the common terms out of the fraction. We start by factoring 2 out of 8 and 14, and then we'll cancel out one \(x\) from the numerator and denominator: \[\frac{8x^2y}{14x} = \frac{2 \cdot 4x^2y}{2 \cdot 7x}\]
03

Simplify the Fraction

Now, cancel the common factors from both the numerator and the denominator. We can cancel \(2\) and \(x\) as shown below:\[\frac{4x^2y}{7x} = \frac{4xy}{7}\]
04

Verify the Simplified Form

To ensure the fraction is simplified correctly, check that there are no further common factors between the numerator and the denominator in \(\frac{4xy}{7}\). As 4, 7, and their variables have no common factors, this is the simplest form.

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Key Concepts

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

Common Factors
When simplifying fractions, identifying common factors plays a key role. A common factor is a number or variable that divides two given numbers or terms exactly. For our fraction \( \frac{8x^2y}{14x} \), both the numerator and the denominator need to be analyzed to find these.
  • Numbers: The numeric part of the numerator is 8, and for the denominator, it is 14. Both of these numbers share a common factor of 2, meaning they can both be divided by 2 without leaving a remainder.
  • Variables: In terms of variables, note the presence of 'x'. Both the numerator and denominator have at least one \(x\). Hence, \(x\) is a common factor here.
Once all common factors are identified, they can be used to simplify the fraction, making it easier to work with or interpret.
Numerator and Denominator
Understanding the role of the numerator and the denominator is fundamental in fraction operations.
  • Numerator: The numerator is the top part of a fraction. In our case, \(8x^2y\) serves as the numerator. It tells us how many 'parts' we have.
  • Denominator: The denominator is the bottom part of a fraction and in our example, it is \(14x\). It indicates the 'whole' or total parts we are dividing into.
In simple terms, when you simplify fractions, you are changing the size of these parts while maintaining the overall value of the fraction. This is done by eliminating common factors from both the numerator and the denominator.
Canceling Terms
Canceling terms is the process of removing common factors from both the numerator and the denominator in order to simplify a fraction.When given the fraction \( \frac{8x^2y}{14x} \), we notice that factors of 2 and \(x\) are common. To simplify:
  • Factor out the common numeric term. Both 8 and 14 have a factor of 2, so write them as \(2 \times 4\) and \(2 \times 7\), respectively.
  • Cancel the common variable \(x\). Each part of the fraction has at least one \(x\), so you can cancel one \(x\) from each, making the remaining part \(\frac{4xy}{7}\).
Canceling terms does not change the value of the fraction overall; it merely presents it in its simplest form, where no further factors can be removed. This not only makes calculations easier but also more visually manageable.

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