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Chapter 11: Problem 55

Express each of the given expressions in simplest form with only positive exponents. $$(D-1)^{-1}+(D+1)^{-1}$$

Short Answer

Expert verified
The expression simplifies to \(\frac{2D}{(D-1)(D+1)}\).

Step by step solution

01

Rewrite with positive exponents

The expression \[(D-1)^{-1} + (D+1)^{-1}\] can be rewritten with positive exponents as \[\frac{1}{D-1} + \frac{1}{D+1}\] because \(x^{-1}\) is equivalent to \(\frac{1}{x}\).
02

Find a common denominator

To add the fractions \(\frac{1}{D-1}\) and \(\frac{1}{D+1}\), we need a common denominator. The common denominator for these fractions is \[(D-1)(D+1),\] which is the product of the two different denominators.
03

Adjust each fraction to have the common denominator

Rewrite each fraction as \[\frac{1}{D-1} = \frac{D+1}{(D-1)(D+1)}\] and \[\frac{1}{D+1} = \frac{D-1}{(D-1)(D+1)}.\] Now both fractions have the same denominator.
04

Add the fractions

Add the fractions \[\frac{D+1}{(D-1)(D+1)} + \frac{D-1}{(D-1)(D+1)}.\] The sum of the numerators is \[(D+1) + (D-1) = 2D.\] Therefore, \[\frac{D+1}{(D-1)(D+1)} + \frac{D-1}{(D-1)(D+1)} = \frac{2D}{(D-1)(D+1)}.\]
05

Simplify the expression

The simplified expression is \[\frac{2D}{(D-1)(D+1)}.\] There are no common factors in the numerator and denominator to cancel, so this is the simplest form.

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Key Concepts

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

Exponents
Exponents are essential components in mathematics that express repeated multiplication of a base number. When dealing with exponents, it's critical to understand how they affect expressions and operations.
  • "Positive exponents" indicate how many times to multiply the base by itself.
  • "Negative exponents" suggest taking the reciprocal of the base raised to the corresponding positive exponent.
When encountering negative exponents, like \(D^{-1}\), converting them to positive exponents, \(\frac{1}{D}\), simplifies the expression. This transformation is key to making complex expressions more manageable, especially when combining terms.
Common Denominator
Finding a common denominator is crucial when adding or subtracting fractions to ensure they have a uniform base for the operation. Here's how you can find a common denominator:
  • Identify the denominators of your fractions.
  • Calculate the least common multiple (LCM) of these denominators.
  • In expressions like \(\frac{1}{D-1} + \frac{1}{D+1}\), multiply the different denominators together to find a common base: \((D-1)(D+1)\).
Once the common denominator is established, each fraction must be adjusted to reflect this base. This adjustment involves multiplying the numerator and denominator by the necessary factors so that all terms share the same denominator.
Fraction Addition
Once fractions share a common denominator, the addition process becomes straightforward:
  • Align each fraction so they have the same denominator.
  • Add the numerators together while maintaining the common denominator.
In the example \(\frac{D+1}{(D-1)(D+1)} + \frac{D-1}{(D-1)(D+1)}\), you simply sum the numerators: \((D+1) + (D-1)\), giving \(2D\). The denominator \((D-1)(D+1)\) remains unchanged. After this operation, always check if you can simplify the expression further, even though the original terms are already in their simplest form.
Rational Expressions
Rational expressions extend the concept of fractions to polynomials in the numerator and denominator. These expressions require careful handling:
  • Simplification often involves reducing the expression to its simplest terms.
  • Factor both the numerator and the denominator to spot any common factors that could be canceled.
With expressions like \((D-1)^{-1} + (D+1)^{-1}\), transforming into positive exponents as fractions allows for further simplification. Post-addition, the expression \frac{2D}{(D-1)(D+1)}\ remains, which is already in its simplest form given no factors cancel further. Mastering rational expressions involves understanding these operations to manage complex algebraic fractions efficiently.

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