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Chapter 6: Problem 97

Perform the indicated operations and simplify. $$ 5(x-3)^{-1}+4(x+3)^{-1}-2(x+3)^{-2} $$

Short Answer

Expert verified
\(\frac{5}{x-3} + \frac{4}{x+3} - \frac{2}{(x+3)^{2}}\)

Step by step solution

01

Distribute Constants

The given expression is:5(x-3)^{-1}+4(x+3)^{-1}-2(x+3)^{-2}Distribute the constants into the terms:\(5 \times (x-3)^{-1}\), \(4 \times (x+3)^{-1}\), and \(-2 \times (x+3)^{-2}\).
02

Identify like terms

Observe the exponents of each term. The expression has terms involving \((x-3)^{-1}\) and \((x+3)^{-1}, (x+3)^{-2}\).
03

Simplify each term

For \((x-3)^{-1}\) and \((x+3)^{-1}\), this means moving them to the denominator. For the term involving \((x+3)^{-2}\), it becomes:\[\frac{5}{x-3} + \frac{4}{x+3} - \frac{2}{(x+3)^{2}}\]

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Key Concepts

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

Distributing Constants
Distributing constants means multiplying each term by a constant value in front of it. In our exercise, the first step was: to distribute constants for the expression:
$$5(x-3)^{-1}+4(x+3)^{-1}-2(x+3)^{-2}$$
To do this, each constant (5, 4, and -2) multiplies the terms they are attached to:
  • 5 multiplies \((x-3)^{-1}\)
  • 4 multiplies \((x+3)^{-1}\)
  • -2 multiplies \((x+3)^{-2}\)
Once distributed, the terms are:
  • \(5 \times (x-3)^{-1}\)
  • \(4 \times (x+3)^{-1}\)
  • \(-2 \times (x+3)^{-2}\)
This operation is simple yet crucial for simplifying expressions.
Identifying Like Terms
Identifying like terms is a key step in simplifying expressions. Like terms have the same variable raised to the same power. In our exercise:
$$5(x-3)^{-1}+4(x+3)^{-1}-2(x+3)^{-2}$$
We need to observe the exponents of each term:
  • \(5(x-3)^{-1}\)
  • \(4(x+3)^{-1}\)
  • \(-2(x+3)^{-2}\)
Here, the like terms are 5(x-3)^{-1} and 4(x+3)^{-1}, both having a \(^{-1}\) exponent. However, \(-2(x+3)^{-2}\) has an exponent of \(^{-2}\), so it is not a like term with the others.
Identifying like terms helps us organize the expression correctly for further simplification.
Simplifying Rational Expressions
Simplifying rational expressions involves reducing the expression to its simplest form. For our given problem, after distributing constants and identifying like terms, the expression needs to be simplified as:
$$\frac{5}{x-3} + \frac{4}{x+3} - \frac{2}{(x+3)^{2}}$$ Each fraction converts its respective term with negative exponents to the denominator. The steps are:
  • 5(x-3)^{-1} becomes \(\frac{5} {x-3}\)
  • 4(x+3)^{-1} becomes \(\frac{4} {x+3}\)
  • -2(x+3)^{-2} becomes \(\frac{-2} {(x+3)^{2}}\)
Combining these simplified terms leads to:
\(\frac {5} {x-3} + \frac {4} {x+3} - \frac {-2} {(x+3)^{2}} \)
Rational expressions can take many forms, and simplifying them steps by spreading out terms, identifying like ones, and reducing them to their lowest terms.

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