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Chapter 6: Problem 9

Multiply. Write each answer in lowest terms. $$ \frac{(x-y)^{2}}{2} \cdot \frac{24}{3(x-y)} $$

Short Answer

Expert verified
The simplified result is \( 4(x-y) \).

Step by step solution

01

Multiply the numerators

Multiply the numerators: \( (x-y)^2 \) and \( 24 \). The result is: \( (x-y)^2 \times 24 = 24(x-y)^{2} \).
02

Multiply the denominators

Multiply the denominators: \( 2 \) and \( 3(x-y) \). The result is: \( 2 \times 3(x-y) = 6(x-y) \).
03

Combine the results

Combine the results from Steps 1 and 2 to form a single fraction: \[ \frac{24(x-y)^{2}}{6(x-y)} \].
04

Simplify the fraction

Simplify the fraction by canceling out the common factor \((x-y)\) in the numerator and the denominator. This results in: \[ \frac{24(x-y)}{6} = \frac{24}{6} \times (x-y) \].
05

Further simplify

Simplify \( \frac{24}{6} \), which equals 4. Thus, the expression \[ 4(x-y) \] is the simplified result.

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Key Concepts

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

Multiplying Fractions
When multiplying fractions, you multiply the numerators together and the denominators together. For instance, given two fractions \(\frac{a}{b} \) and \(\frac{c}{d}\), multiply the numerators (a \cdot c) and the denominators (b \cdot d). This gives you the product of the fractions: \(\frac{a \cdot c}{b \cdot d}\). This step is straightforward, you just need to be careful with any variables or expressions involved, like in our example where we had \((x-y)^2\) and 24 in the numerators, and 2 and (3(x-y)) in the denominators.
Simplifying Expressions
Simplifying expressions is crucial for making complicated math problems easier to manage. When you combine like terms or factor expressions, you are simplifying. In the exercise solution, we got to \(\frac{24(x-y)^2}{6(x-y)}\). To simplify, you need to cancel out common factors. Just focus on identifying and reducing them. Carrying out the operation, we cancel \((x-y)\) common to both the numerator and the denominator. This reduces our expression to \(\frac{24(x-y)}{6}\).
Canceling Common Factors
Canceling common factors helps to simplify fractions to their lowest terms. To cancel, identify the factors that are the same in both the numerator and the denominator. Once identified, they can be eliminated (effectively divided out). For instance, in our exercise, \((x-y)\) is present in both the numerator and the denominator. By canceling \((x-y)\), the fraction reduces to \(\frac{24}{6} \cdot (x-y) \). Finally, simplifying \(\frac{24}{6}\) gives us 4. Therefore, the final simplified product is 4(x-y).

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Most popular questions from this chapter

involve factoring by grouping (Section 5.1) and factoring sums and differences of cubes (Section 5.4). Write each rational expression in lowest terms $$ \frac{m^{2}-n^{2}-4 m-4 n}{2 m-2 n-8} $$

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Multiply or divide as indicated. Write each answer in lowest terms. $$ \frac{3}{7} \cdot \frac{4}{5} $$
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