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Chapter 6: Problem 60

Simplify each complex fraction. Assume no division by 0. $$\frac{1+x^{-1}}{x^{-1}-1}$$

Short Answer

Expert verified
The simplified form of the given complex fraction is \((x+1)/(1-x)\).

Step by step solution

01

Step 1:Convert Negative Exponents into Positive

The negative exponents in this fraction can be rewritten as positive exponents by taking their reciprocals. So, \(x^{-1}\) can be rewritten as \(1/x\). So, the fraction becomes \(\frac{1+(1/x)}{(1/x)-1}\).
02

Simplify the Numerator

In order to omit the fractions inside the fraction, we can rewrite the numerator \(1+(1/x)\) as \((x+1)/x\).
03

Step 3:Simplify the Denominator

Similarly, the denominator \((1/x)-1\) can be rewritten as \(1-x/x\).
04

Simplify the Complex Fraction

Now, the complex fraction becomes \(\frac{(x+1)/x}{(1-x)/x}\). Since the fractions \((x+1)/x\) and \((1-x)/x\) have the same denominator, they can be rewritten as \((x+1)/(1-x)\). Therefore, simplifying the complex fraction yields \(x+1)/(1-x)\).

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Key Concepts

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

Negative Exponents
Negative exponents can often seem puzzling, but they're quite simple once you understand the concept. When you see a negative exponent, like in the term \(x^{-1}\), it indicates the reciprocal of that base raised to the corresponding positive exponent. You can think of it as flipping the number over. For example, \(x^{-1}\) becomes \(1/x\). This is a powerful tool for rewriting expressions in a more manageable form, especially when dealing with complex fractions.
Reciprocals
Reciprocals are a key concept in mathematics, especially when simplifying expressions and solving equations. The reciprocal of a number is what you multiply it by to get the product of 1. For example, the reciprocal of \(x\) is \(1/x\). In terms of negative exponents, we convert them into reciprocals to simplify fractions. Hence, \(x^{-1} = 1/x\), transforming complex expressions into simpler forms without changing their value.
Numerator and Denominator Simplification
Simplifying the numerator and denominator individually can greatly ease the process of dealing with complex fractions. By converting terms with negative exponents into their reciprocal fractions, you can combine different parts more easily.
In the given exercise, the numerator \(1 + (1/x)\) is converted to \((x+1)/x\) by finding a common denominator of \(x\), allowing for direct addition. Similarly, the denominator \((1/x) - 1\) becomes \((1-x)/x\). This prepares each part for easier manipulation and further simplification.
Simplifying Fractions
The ultimate goal of dealing with fractions is to simplify them as much as possible. This involves reducing them to their simplest form while maintaining equivalence to the original expression. For complex fractions, once the numerator and denominator have the same denominator, simplification is much more straightforward.
In the exercise, after expressing both the numerator and denominator with \(x\) as a common denominator, the fraction simplifies beautifully to \((x+1)/(1-x)\). This expression no longer contains complex fractions within fractions, making it easier to interpret and work with in further calculations.

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