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Chapter 7: Problem 77

perform the indicated operation and simplify. $$\frac{2}{x} \cdot \frac{8}{y}$$

Short Answer

Expert verified
\( \frac{16}{xy} \)

Step by step solution

01

Multiply the Numerators

First, multiply the numerators of the fractions together. Here, the numerators are 2 and 8. So, multiply 2 by 8 which gives you 16.
02

Multiply the Denominators

Next, multiply the denominators of the fractions together. The denominators are x and y. Thus, multiply x by y to get xy.
03

Form the New Fraction

Combine the results from Steps 1 and 2 to form the new fraction. This will give you the fraction: \[ \frac{16}{xy} \]
04

Simplify (if possible)

Check if the fraction \[ \frac{16}{xy} \] can be simplified further. Since 16 and xy have no common factors, this is already in its simplest form.

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Key Concepts

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

Numerators
When multiplying fractions, the numerator is the number on the top part of a fraction. It represents how many parts of a whole you have. In the exercise, the fractions are \(\frac{2}{x}\) and \(\frac{8}{y}\). The numerators are 2 and 8.
To multiply the numerators, simply multiply these top numbers together. So, 2 multiplied by 8 gives us 16. This is the new numerator of the product.
Here's a recap:
  • Identify the numerators: 2 and 8.
  • Multiply them: 2 * 8 = 16.
  • This result, 16, becomes the numerator of the new fraction.
Denominators
The denominator is the bottom part of a fraction. It shows the total number of equal parts the whole is divided into. In our fractions \(\frac{2}{x}\) and \(\frac{8}{y}\), the denominators are x and y.
To multiply the denominators:
  • First identify the denominators: x and y.
  • Then multiply them: x * y = xy.
This gives us the new denominator for the product of the fractions. So, for our problem, the new denominator is xy.
Here's what we've done:
  • Identified the denominators: x and y.
  • Multiplied them: x * y = xy.
  • This product, xy, is the new denominator of our resulting fraction.
Simplification
Simplification is the process of reducing a fraction to its simplest form. To simplify, we check if there are any common factors between the numerator and the denominator. If any, we divide both by the greatest common divisor.
In our case, the result from multiplying the fractions is \(\frac{16}{xy}\). We need to see if 16 and xy have any common factors. Here:
  • The numerator is 16.
  • The denominator is xy, which consists of x and y.
The numbers 16, and the variables x and y, have no common factors. So, \(\frac{16}{xy}\) is already in its simplest form.
To summarize:
  • Identify common factors between the numerator and denominator.
  • If no common factors exist, the fraction is already simplified.

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