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Chapter 0: Problem 47

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

Short Answer

Expert verified
The simplified expression for the given complex fraction is \(\dfrac{x^2-4}{yz}\).

Step by step solution

01

Invert the divisor

The given expression is a complex fraction of the form \(\dfrac{\dfrac{x-2}{y}}{\dfrac{z}{x+2}}\), where \(\dfrac{x-2}{y}\) is the dividend and \(\dfrac{z}{x+2}\) is the divisor. We need to invert the divisor, which means swapping the numerator and denominator of \(\dfrac{z}{x+2}\). The inverted divisor will be \(\dfrac{x+2}{z}\).
02

Multiply the dividend by the inverted divisor

Now, we need to multiply the dividend (\(\dfrac{x-2}{y}\)) by the inverted divisor (\(\dfrac{x+2}{z}\)). We can do this by multiplying the numerators together and the denominators together: \((\dfrac{x-2}{y})(\dfrac{x+2}{z}) = \dfrac{(x-2)(x+2)}{yz}\)
03

Simplify the result

In this step, simplify the multiplication in the numerator: \((x-2)(x+2)\) is a difference of two squares, which can be simplified to \(x^2 - 2^2\), or \(x^2 - 4\). Therefore, the simplified fraction is: \(\dfrac{x^2-4}{yz}\) The simplified expression for the given complex fraction is \(\dfrac{x^2-4}{yz}\).

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Key Concepts

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

Difference of Squares
The concept of the difference of squares is crucial when simplifying certain expressions. It applies when you have an expression of the form \((a-b)(a+b)\) which can be rewritten as \(a^2 - b^2\). This is one of the fundamental algebraic identities.
This formula helps to quickly simplify expressions by recognizing this pattern. In our complex fraction, the expression \((x-2)(x+2)\) represents a classic difference of squares. Here, \(a = x\) and \(b = 2\). So, applying the formula gives us \(x^2 - 4\).
Remembering this identity can save a lot of time and simplify many algebraic computations. You'll often encounter it in problems involving polynomials and fractions.
Fraction Simplification
Fraction simplification is the process of reducing fractions to their simplest form. In this context, it involves ensuring that the numerator and denominator have no common factors other than 1.
To simplify a complex fraction, like the one we have, you follow a straightforward approach:
  • Multiply straight across, combining the numerators and denominators as identified.
  • Use algebraic identities, like the difference of squares, to simplify.
In our fraction, after inverting the divisor and multiplying, you get: \(\dfrac{(x-2)(x+2)}{yz}\). Utilizing the difference of squares for the numerator, it simplifies to \(\dfrac{x^2 - 4}{yz}\).
Always check your final result to confirm there are no further common factors. This ensures the fraction is in its simplest possible form.
Numerator and Denominator Inversion
In complex fractions, inversion is a key step to simplify expressions. The rule here is to invert the divisor. This means swapping the numerator and the denominator.
When you see a fraction over another fraction, like \(\dfrac{\dfrac{x-2}{y}}{\dfrac{z}{x+2}}\), you take the fraction in the denominator and flip it upside down.
Applying this to our exercise, we have:
  • Original divisor: \(\dfrac{z}{x+2}\)
  • Inverted divisor: \(\dfrac{x+2}{z}\)
Once inverted, you multiply it with the numerator of the complex fraction. This transformation is crucial because it converts the division of fractions into a multiplication problem, which is easier to handle.
Understanding this technique can greatly aid in manipulating and simplifying fractions efficiently.

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Most popular questions from this chapter

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

Find all numbers \(x\) satisfying the given equation. $$|x+3|=x+5$$

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

The sales tax in San Francisco is \(8.5 \% .\) Diners in San Francisco often compute a \(17 \%\) tip on their beforetax 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 four examples of pairs of real numbers \(a\) and \(b\) such that \(|a+b|=3\) and \(|a|+|b|=11\).
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