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Chapter 6: Problem 23

Simplify each complex fraction. Use either method. $$ \frac{a-\frac{5}{a}}{a+\frac{1}{a}} $$

Short Answer

Expert verified
\( \frac{a^2 - 5}{a^2 + 1} \)

Step by step solution

01

- Simplify the Numerator

Combine the terms in the numerator by finding a common denominator. The common denominator for the terms in the numerator is 鈥榓鈥�. So, rewrite the numerator as follows: \[ a - \frac{5}{a} = \frac{a^2}{a} - \frac{5}{a} = \frac{a^2 - 5}{a} \]
02

- Simplify the Denominator

Combine the terms in the denominator by finding a common denominator. The common denominator for the terms in the denominator is also 鈥榓鈥�. So, rewrite the denominator as follows: \[ a + \frac{1}{a} = \frac{a^2}{a} + \frac{1}{a} = \frac{a^2 + 1}{a} \]
03

- Form the Complex Fraction

Combine the simplified numerator and denominator into a single fraction with division: \[ \frac{\frac{a^2 - 5}{a}}{\frac{a^2 + 1}{a}} \]
04

- Simplify the Complex Fraction

Simplify the complex fraction by multiplying the numerator by the reciprocal of the denominator: \[ \frac{a^2 - 5}{a} \div \frac{a^2 + 1}{a} = \frac{a^2 - 5}{a} \times \frac{a}{a^2 + 1} = \frac{a^2 - 5}{a^2 + 1} \]

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Key Concepts

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

Common Denominator
In fractions, a common denominator is a shared multiple of the denominators of several fractions. When adding or subtracting fractions, we need a common denominator to make the fractions comparable and perform the arithmetic operations conveniently. In the exercise, for the terms in the numerator which are \(a\) and \(\frac{5}{a}\), the common denominator is identified as \(a\). Similarly, for the terms in the denominator \(a\) and \(\frac{1}{a}\), the common denominator is again \(a\).
By writing both terms with the same denominator, we can easily combine them into a single fraction and then proceed with further operations.
Algebraic Fractions
Algebraic fractions are fractions where the numerator and/or the denominator contain algebraic expressions. Simplifying algebraic fractions involves dealing with expressions in terms of variables. This requires knowledge of algebraic rules and operations.
In our example, the fractions \(\frac{5}{a}\) and \(\frac{1}{a}\) are algebraic fractions. When simplifying algebraic fractions, it's vital to combine terms with a common denominator and then simplify further as needed. Here, we combined them to get \( \frac{a^2 - 5}{a} \) and \( \frac{a^2 + 1}{a} \).
Fraction Division
Fraction division involves dividing one fraction by another. When dividing fractions, we multiply by the reciprocal of the divisor. The reciprocal of a fraction is formed by swapping its numerator and denominator.
In our step-by-step solution, we encountered the complex fraction \( \frac{ \frac{a^2 - 5}{a}}{\frac{a^2 + 1}{a}} \). To divide these fractions, we converted the division into multiplication by the reciprocal. This means we turned \(\frac{a^2 + 1}{a}\) into its reciprocal \(\frac{a}{a^2 + 1}\) and then multiplied.
Reciprocal Multiplication
Reciprocal multiplication simplifies the process of division by allowing multiplication in place of division. The reciprocal of a fraction \(\frac{p}{q}\) is \(\frac{q}{p}\). When we multiply by the reciprocal, we effectively divide.
In our context, the final step involved multiplying \( \frac{a^2 - 5}{a} \) by the reciprocal of \( \frac{a^2 + 1}{a} \). This resulted in multiplication: \( \frac{a^2 - 5}{a} \times \frac{a}{a^2 + 1} \). After performing this multiplication, we simplified to get \( \frac{a^2 - 5}{a^2 + 1} \).
Using this method ensures correct and simplified results effectively.

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Most popular questions from this chapter

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.

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

Simplify each complex fraction. Use either method. $$ \frac{2+\frac{1}{x}-\frac{28}{x^{2}}}{3+\frac{13}{x}+\frac{4}{x^{2}}} $$

Simplify each complex fraction. Use either method. $$ \frac{6+\frac{2}{r}}{\frac{3 r+1}{4}} $$

Simplify each expression, using only positive exponents in the answer. $$ \frac{x^{-1}+2 y^{-1}}{2 y+4 x} $$
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