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Chapter 7: Problem 19

\(\frac{\left(\frac{x}{4}-\frac{4}{x}\right)}{(x-4)}\)

Short Answer

Expert verified
The simplified form of the complex fraction \(\frac{\left(\frac{x}{4}-\frac{4}{x}\right)}{(x-4)}\) is \((x+4) / 4x\)

Step by step solution

01

Breaking down the complex fraction

A complex fraction can be broken down by multiplying the numerator and denominator by the least common multiple (LCM) of the denominators in the complex fraction. In this case, the LCM of 4 and \(x\) is \(4x\). So, the fraction becomes: \(4x 脳 \frac{x}{4} - 4x 脳 \frac{4}{x}\) / \(4x 脳 (x-4)\)
02

Simplify the equation

After multiplying the fractions in the numerator and denominator by the LCM, simplify the resulting fractions. This leads to: \(x^2 - 16\) / \(4x^2 - 16x\)
03

Factor out the numerator and denominator

Factor out the numerator and denominator to simplify them even further. This results in: \((x+4)(x-4)\) / \(4x(x-4)\)
04

Cancel out common factors

The terms \((x - 4)\) in the numerator and denominator cancel each other out. Therefore, the fraction simplifies to: \((x+4) / 4x\)

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Key Concepts

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

Least Common Multiple (LCM)
When dealing with complex fractions, finding the least common multiple (LCM) is a useful strategy. The LCM is the smallest number that is a multiple of the denominators. This process helps in reducing complex fractions to simpler forms by getting a common base for each term.

For example, if you have denominators 4 and \(x\), you need to find a number that both can divide evenly. The LCM is \(4x\), since 4 can divide 4 once and \(x\) divides \(x\) once as well. Multiplying by the LCM allows us to eliminate the fractions within the main fraction, making it easier to simplify.
Factoring Polynomials
Factoring is a technique used to express a polynomial as a product of its factors. This is essential for further simplification of algebraic expressions, especially when solving complex fractions. Factoring helps in revealing common factors that can be canceled or further simplified.

For the expression \(x^2 - 16\), we notice it's a difference of squares. This can be factored into \((x+4)(x-4)\). Similarly, the expression \(4x^2 - 16x\) can be factored by taking out a common factor, in this case, 4x, leading to \(4x(x-4)\). Factoring makes it clear which elements of the expressions can cancel out in the simplification process.
Canceling Common Factors
Once a polynomial is factored, it is often possible to simplify it further by canceling out common factors from both the numerator and the denominator. This process helps in reducing the complexity of the fraction.

In our factored expression \((x+4)(x-4) / 4x(x-4)\), we see that \((x-4)\) is present in both the numerator and the denominator. By canceling these common factors, the expression simplifies to \((x+4) / 4x\).
  • Always ensure to only cancel factors that are common and occur in both parts.
  • Watch out for factors in the denominator that could potentially make the denominator zero, as these are restrictions (in this case, \(x eq 4\)).

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Most popular questions from this chapter

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