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Chapter 0: Problem 49

Simplify the given expression possible. $$ \frac{\frac{x-2}{y}}{\frac{z}{x+2}} $$

Short Answer

Expert verified
Our final simplified expression is \(\frac{(x-2)(x+2)}{yz}\).

Step by step solution

01

Rewrite the expression

Rewrite the given expression as a multiplication problem by multiplying the first fraction by the reciprocal of the second fraction: $$ \frac{\frac{x-2}{y}}{\frac{z}{x+2}} = \frac{x-2}{y} \cdot \frac{x+2}{z} $$
02

Simplify the numerator and denominator

In this case, there isn't anything we can simplify within the numerator and denominator since there are no common factors to cancel out. So, we can skip this step.
03

Multiply the simplified fractions

Now, we multiply the two fractions: $$ \frac{x-2}{y} \cdot \frac{x+2}{z} = \frac{(x-2)(x+2)}{yz} $$ Our final simplified expression is: $$ \frac{(x-2)(x+2)}{yz} $$

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Key Concepts

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

Reciprocal of a Fraction
Understanding the reciprocal of a fraction is crucial when working with complex fractions, such as in our exercise. The reciprocal of a fraction is simply obtained by flipping its numerator and denominator. For example, the reciprocal of \( \frac{a}{b} \) is \( \frac{b}{a} \) provided that neither \( a \) nor \( b \) is zero.

In the exercise provided, we used the reciprocal to transform a division problem into a multiplication problem. The reciprocal of \( \frac{z}{x+2} \) is \( \frac{x+2}{z} \) and multiplying by this reciprocal effectively 'cancels out' the division by the second fraction, simplifying our work.
Simplifying Fractions
Simplifying fractions is all about making them easier to work with, ideally reducing them to their simplest form where the numerator and denominator share no common factors besides 1. To simplify a fraction, we look for common factors in the numerator and the denominator that can be divided out.

However, in our step-by-step solution we observed that the numerator \( x-2 \) and denominator \( y \) of the first fraction, and the numerator \( x+2 \) and denominator \( z \) of the second fraction did not have any factors in common. So, no simplification was possible beyond multiplying the fractions.
Multiplying Fractions
When it comes to multiplying fractions, the process is straight-forward: we multiply the numerators together to get the new numerator and the denominators together to get the new denominator.

In our example, we multiplied the numerators \( x-2 \) and \( x+2 \) to obtain \( (x-2)(x+2) \) and we multiplied the denominators \( y \) and \( z \) to get \( yz \) for our final expression \( \frac{(x-2)(x+2)}{yz} \). Multiplication of fractions is usually easier than simplifying complex fractions, as it doesn't usually require finding common denominators.
Numerator and Denominator
The numerator and denominator are the two parts of a fraction, separated by the fraction bar. The numerator, located above the bar, represents the number of parts we're considering, while the denominator, below the bar, indicates the total number of equal parts that make up a whole.

In the context of our exercise, \( x-2 \) and \( (x-2)(x+2) \) serve as numerators in different steps of the solution, while \( y \) and \( yz \) serve as denominators. Understanding how to work with numerators and denominators is essential to perform operations such as simplification or multiplication of fractions.

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Most popular questions from this chapter

Suppose \(t\) is an irrational number. Explain why at least one of \(t^{2}\) and \(t^{3}\) is irrational.

The intersection of two sets of numbers consists bers that are in both sets. If \(A\) and \(B\) are sets, intersection is denoted by \(A \cap B\). Write each intersection as a single interval. $$ [-2,8] \cap(-1,4) $$

Explain why $$ |a b|=|a||b| $$ for all real numbers \(a\) and \(b\).

Write each set as an interval or of two intervals. $$ \left\\{x:|4 x-3|<\frac{1}{5}\right\\} $$

Write each union as a single interval. $$ (-9,-2) \cup[-7,-5] $$
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