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Chapter 6: Problem 53

Simplify each complex fraction. Use either method. $$ \frac{\frac{1}{x^{3}-y^{3}}}{\frac{1}{x^{2}-y^{2}}} $$

Short Answer

Expert verified
\(\frac{x+y}{x^{2} + xy + y^{2}}\)

Step by step solution

01

- Understand the Complex Fraction

A complex fraction has a fraction in its numerator, its denominator, or both. Here, we have a fraction: \(\frac{\frac{1}{x^{3} - y^{3}}}{\frac{1}{x^{2} - y^{2}}}\). We will simplify it by dealing with the fractions in the numerator and the denominator separately.
02

- Rewrite the Complex Fraction

Rewrite the given expression as a division of two fractions: \(\frac{1}{x^{3} - y^{3}} \times \frac{x^{2} - y^{2}}{1}\). This turns the complex fraction into a multiplication of a fraction and its reciprocal.
03

- Simplify the Multiplication

When multiplying fractions, multiply the numerators together and the denominators together: \(\frac{1 \times (x^{2} - y^{2})}{(x^{3} - y^{3}) \times 1}\). The expression simplifies to: \(\frac{x^{2} - y^{2}}{x^{3} - y^{3}}\).
04

- Factor Both the Numerator and the Denominator

Factor the numerator \(x^{2} - y^{2}\) as a difference of squares: \((x+y)(x-y)\). Factor the denominator \(x^{3} - y^{3}\) as a difference of cubes: \((x-y)(x^{2} + xy + y^{2})\). This gives us: \(\frac{(x+y)(x-y)}{(x-y)(x^{2} + xy + y^{2})}\).
05

- Cancel Common Factors

Cancel the common factor \(x - y\) in the numerator and the denominator: \(\frac{(x+y)(x-y)}{(x-y)(x^{2} + xy + y^{2})} = \frac{(x+y)}{(x^{2} + xy + y^{2})}\).
06

- Write the Final Simplified Form

The expression simplifies to: \(\frac{x+y}{x^{2} + xy + y^{2}}\). Hence, the complex fraction is fully simplified.

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Key Concepts

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

Difference of Squares
A difference of squares is a specific type of polynomial that takes the form:
  • \(a^2 - b^2 = (a + b)(a - b)\)
This pattern is useful because it helps to break down expressions into simpler factors. Let's use an example: Suppose we have \(x^2 - y^2\). It can be factored as \(x^2 - y^2 = (x + y)(x - y)\). This is just what we did in the exercise. We took \(x^2 - y^2\) and rewrote it as \((x + y)(x - y)\). Using this identity makes it easier to simplify or solve equations because you are working with smaller, more manageable terms.
Difference of Cubes
The difference of cubes is another factoring pattern but works specifically with cubes. It is written as:
  • \(a^3 - b^3 = (a - b)(a^2 + ab + b^2)\)
In our exercise, we factored \(x^3 - y^3\): We rewrite \(x^3 - y^3\) as \((x - y)(x^2 + xy + y^2)\). This helps simplify the expression because it separates the original polynomial into smaller components. By breaking down complex expressions, you make difficult-looking problems much simpler to handle. Just remember this key identity the next time you see a perfect cube:
  • \(x^3 - y^3\) can always be rewritten as \((x - y)(x^2 + xy + y^2)\).
Fraction Multiplication
Multiplying fractions might seem challenging at first, but it's straightforward once you get the hang of it. Here's how you do it:
  • Multiply the numerators together.
  • Multiply the denominators together.
That鈥檚 it! In the exercise, we converted the complex fraction into multiplication: \(\frac{1}{x^3 - y^3} \times \frac{x^2 - y^2}{1}\). This resulted in: \(\frac{1 \times (x^2 - y^2)}{(x^3 - y^3} \times 1)\), leading to \(\frac{x^2 - y^2}{x^3 - y^3}\). Simplifying multiplication of fractions makes it possible to factor and cancel common terms easily. When multiplying fractions:
  • Always check if you can simplify any terms first.
  • Multiply straight across鈥攏umerator with numerator and denominator with denominator.
  • Simplify the resulting fraction, if possible.
By following these steps, complex fractions become less daunting. Practice makes perfect, so try more problems to get comfortable with these techniques.

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Most popular questions from this chapter

Find the slope of the line that passes through each pair of points. This will involve simplifying complex fractions. $$ \left(\frac{1}{2}, \frac{5}{12}\right) \text { and }\left(\frac{1}{4}, \frac{1}{3}\right) $$

Simplify each complex fraction. Use either method. $$ \frac{\frac{6}{5}-\frac{1}{9}}{\frac{2}{5}+\frac{5}{3}} $$

A Concours d'Elegance is a competition in which a maximum of 100 points is awarded to a car based on its general attractiveness. The rational expression $$ \frac{9010}{49(101-x)}-\frac{10}{49} $$ approximates the cost, in thousands of dollars, of restoring a car so that it will win x points. 101\. Simplify the given expression by performing the indicated subtraction.

Solve each formula or equation for the specified variable. $$ \frac{5}{y+2}-\frac{r}{y-2}=\frac{3}{y^{2}-4} \text { for } r $$

In these subtraction problems, the rational expression that follows the subtraction sign has a numerator with more than one term. Be careful with signs and find each difference. See Examples \(6-10 .\) 76\. \(\frac{1}{a^{2}-1}-\frac{a-1}{a^{2}+3 a-4}\)
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