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Chapter 6: Problem 11

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

Short Answer

Expert verified
4(x-y)

Step by step solution

01

- Factorize the Numerator

Factorize any numerators completely if possible: The numerator \(24\) on the right fraction is already in its factored form. The left fraction's numerator \((x-y)^2\) is already in its simplest form.
02

- Combine the Fractions

Rewrite the expression as a single fraction: \[ \frac{(x-y)^2}{2} \cdot \frac{24}{3(x-y)} = \frac{(x-y)^2 \cdot 24}{2 \cdot 3(x-y)} \]
03

- Simplify

Simplify the expression by canceling out common factors: \[ \frac{(x-y)^2 \cdot 24}{2 \cdot 3(x-y)} = \frac{(x-y) \cdot 24}{2 \cdot 3} \]
04

- Further Simplification

Simplify the remaining fraction: \[ \frac{(x-y) \cdot 24}{2 \cdot 3} = \frac{(x-y) \cdot 24}{6} \] \ \[ = (x-y) \cdot 4 \] \ \[ = 4(x-y) \]

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Key Concepts

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

Factoring Expressions
Factoring is breaking down an expression into simpler parts that, when multiplied together, give you the original expression. It's like finding the pieces of a puzzle that fit perfectly to form a larger picture.
In our exercise, the numerator \((x-y)^2\) is already in its simplest form because it is a perfect square. It's like packing a suitcase tightly without leaving any extra space.
For more complex expressions, you might need to use different factoring techniques such as using the distributive property or recognizing special products like the difference of squares or the sum of cubes. If an expression can not be factored, it is considered
Simplifying Fractions
Simplifying fractions means making the fraction as simple as possible.
This is done by eliminating any common factors in the numerator and the denominator.
To simplify our example, we first recognized that \((x-y)^2\) and \(3(x-y)\) have the common factor \(x-y\).
We cancelled out \(x-y\) from both the numerator and the denominator.
When you simplify fractions, you often look for the greatest common divisor (GCD) of the numerator and the denominator. This helps reduce the fraction to its lowest terms. Imagine you're sharing a pizza; you want everyone to get the simplest, most equal portion possible.
Fraction Multiplication
Multiplying fractions involves multiplying the numerators together and the denominators together.
There's no need to find a common denominator for multiplication, making it a straightforward process.
In our exercise, we rewrote the expression as a single fraction and then multiplied the numerators \((x-y)^2 \cdot 24\) and the denominators \(2 \cdot 3 (x-y)\).
After multiplying, we had to simplify by canceling out common factors.
Think of multiplying fractions like stacking Lego blocks; each block (or number) builds on top of the other. After building, you sometimes have to remove extra pieces to make your structure neat and clean.

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Most popular questions from this chapter

The fractions here are continued fractions. Simplify by starting at "the bottom" and working upward. $$ 1+\frac{1}{1+\frac{1}{1+1}} $$

Solve each formula or equation for the specified variable. $$ h=\frac{2 s \ell}{B+b} \text { for } B $$

Find the slope of the line that passes through each pair of points. This will involve simplifying complex fractions. $$ \left(-\frac{2}{9}, \frac{5}{18}\right) \text { and }\left(\frac{1}{18},-\frac{5}{9}\right) $$

Simplify each complex fraction. Use either method. $$ \frac{\frac{1}{x^{2}}+\frac{1}{y^{2}}}{\frac{1}{x}-\frac{1}{y}} $$

Simplify each expression, using only positive exponents in the answer. $$ \frac{k^{-1}+p^{-2}}{k^{-1}-3 p^{-2}} $$
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