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Chapter 0: Problem 50

Simplify the given expression possible. $$ \frac{\frac{x-4}{y+3}}{\frac{y-3}{x+4}} $$

Short Answer

Expert verified
The short version of the answer is: \[ \frac{(x-4)(x+4)}{(y+3)(y-3)} \]

Step by step solution

01

Identify the given expression

Here, we are asked to simplify the following expression: \[ \frac{\frac{x-4}{y+3}}{\frac{y-3}{x+4}} \]
02

Apply the rule for dividing fractions

To divide fractions, we will flip the second fraction and convert the division operation to multiplication. Therefore, the given expression can be rewritten as: \[ \frac{x-4}{y+3} \times \frac{x+4}{y-3} \]
03

Simplify the expression

Notice that there are no common factors in both the numerator and the denominator, so we simply multiply the numerators together and the denominators together: \[ \frac{(x-4)(x+4)}{(y+3)(y-3)} \]
04

Final result

After all the steps, the simplified expression is: \[ \frac{(x-4)(x+4)}{(y+3)(y-3)} \]

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Key Concepts

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

Dividing Fractions
Dividing fractions is a fundamental concept in math that involves changing the way we look at fractions. When dividing one fraction by another, the problem can be transformed into a multiplication problem. You achieve this by flipping or "taking the reciprocal" of the second fraction and then multiplying the numerators and denominators of the fractions. For example, in the given exercise:
  • Original division: \( \frac{\frac{x-4}{y+3}}{\frac{y-3}{x+4}} \)
  • Rewrite as multiplication: \( \frac{x-4}{y+3} \times \frac{x+4}{y-3} \)
This method simplifies solving fraction division issues, as multiplication is often conceptually less challenging than division for fractions. By flipping the second fraction and multiplying instead, you avoid direct division while still achieving the same result.
Numerator and Denominator
Fractions consist of two essential parts: the numerator and the denominator. The numerator is located at the top of the fraction and tells us how many parts we are considering. The denominator, on the bottom, shows the total number of equal parts the whole is divided into.
  • In \( \frac{3}{4} \), the numerator is 3, meaning three parts are taken.
  • The denominator is 4, indicating that the whole is divided into four parts.
In the simplification of the expression \( \frac{\frac{x-4}{y+3}}{\frac{y-3}{x+4}} \), each component fraction has its numerator and denominator, which must be multiplied across when using the multiplication method. As we multiply, we get a new numerator \((x-4)(x+4)\) and a new denominator \((y+3)(y-3)\). Recognizing and handling these parts correctly is crucial in both understanding and simplifying fractions.
Rational Expressions
Rational expressions are expressions that involve fractions with polynomials in the numerator and the denominator. Simplifying rational expressions follows similar rules to numerical fractions, where we find common factors and reduce them to their simplest form if possible.In our expression \( \frac{(x-4)(x+4)}{(y+3)(y-3)} \), it's crucial to understand that though it's already simplified, identifying factors is key:
  • Numerator: Expression \((x-4)(x+4)\) can resemble a difference of squares, usually \((a-b)(a+b)\) which is \(a^2-b^2\).
  • Denominator: \((y+3)(y-3)\) also resembles a difference of squares.
The idea with rational expressions is to simplify where common factors appear in the numerator and the denominator. In this particular case, there's no further simplification possible, showing how understanding the structures helps avoid unnecessary errors.

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Most popular questions from this chapter

A quick way to compute a \(15 \%\) tip on a restaurant first to compute \(10 \%\) of the bill (by shifting the de point) and then add half of that amount for the to For example, \(15 \%\) of a \(\$ 43\) restaurant bill is \(\$ 4.30+\) which equals \$6.45. Explain why this technique application of the distributive property.

Simplify the given expression possible. $$ \frac{\frac{1}{x+a}-\frac{1}{x}}{a} $$

The sales tax in San Francisco is \(8.5 \%\). Diners in San Francisco often compute a \(17 \%\) tip on their before-tax restaurant bill by simply doubling the sales tax. For example, a \(\$ 64\) dollar food and drink bill would come with a sales tax of \(\$ 5.44\); doubling that amount would lead to a \(17 \%\) tip of \(\$ 10.88\) (which might be rounded up to \$11). Explain why this technique is an application of the associativity of multiplication.

Give an example of two irrational numbers whose product is an irrational number.

Find all numbers \(x\) satisfying the given inequality. $$ \left|\frac{4 x+1}{x+3}\right|<2 $$
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