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Chapter 5: Problem 131

Simplify each expression. (a) \(7 n^{-1}\) (b) \((7 n)^{-1}\) (c) \((-7 n)^{-1}\)

Short Answer

Expert verified
(a) \(\frac{7}{n}\), (b) \(\frac{1}{7n}\), (c) \(-\frac{1}{7n}\)

Step by step solution

01

Simplify Part (a)

For the expression \(7 n^{-1}\), recall that a negative exponent means we take the reciprocal of the base. So, \(n^{-1}\) is the same as \(\frac{1}{n}\). Thus, the expression simplifies to \(\frac{7}{n}\).
02

Simplify Part (b)

For the expression \((7 n)^{-1}\), again use the reciprocal property of a negative exponent. This time the base is \(7n\), so we take the reciprocal to get \(\frac{1}{7n}\).
03

Simplify Part (c)

For \((-7 n)^{-1}\), use the same property. The base here is \(-7n\), so the reciprocal is \(\frac{1}{-7n}\), which simplifies to \(-\frac{1}{7n}\).

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Key Concepts

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

Reciprocal
The concept of the reciprocal is vital when dealing with negative exponents. Essentially, the reciprocal of a number is 1 divided by that number. For example:
    \t
  • The reciprocal of 2 is \(\frac{1}{2}\).
    • \t
  • The reciprocal of \(\frac{5}{3}\) is \(\frac{3}{5}\).
    • \t
  • The reciprocal of variable \(\frac{a}{b}\) is \(\frac{b}{a}\).
When we encounter a negative exponent in an expression, it indicates that we should use the reciprocal of the base. For instance, in part (a) of the exercise, the expression \(n^{-1}\) can be interpreted as taking the reciprocal of \(n\), resulting in \(\frac{1}{n}\). Therefore, \(7 n^{-1}\) becomes \(\frac{7}{n}\).
Simplifying Expressions
Simplification makes mathematical expressions easier to work with. It involves reducing an expression to its simplest form, removing any unnecessary complexity. Here鈥檚 how you can simplify expressions containing negative exponents:
    \t
  • Identify and separate components with negative exponents.
    • \t
  • Take the reciprocal of the base of each negative exponent.
    • \t
  • Rewrite the expression using positive exponents.
Take part (b) of the exercise: \( (7n)^{-1} \). The negative exponent tells us to take the reciprocal of the entire base 7n. Hence, \((7n)^{-1}\) becomes \(\frac{1}{7n}\). Similarly, simplifying \((-7n)^{-1}\) in part (c) results in \(\frac{1}{-7n}\), which is \(-\frac{1}{7n}\). Using these methods, all expressions become more manageable.
Negative Exponent Rules
Understanding negative exponent rules is crucial for simplifying expressions. The essential rule states: \(a^{-n} = \frac{1}{a^n}\). This rule indicates two key points:
    \t
  • A negative exponent signifies the reciprocal of the base raised to the opposite positive exponent.
    • \t
  • Always flip the base and change the sign of the exponent.
For example:
    \t
  • \(2^{-3} = \frac{1}{2^3} = \frac{1}{8}\)
    • \t
  • \(x^{-4} = \frac{1}{x^4}\)
    • \t
  • \(\frac{1}{a^{-3}} = a^3\)
Using these rules, part (b) of the exercise, \( (7n)^{-1}\), simplifies to \(\frac{1}{7n}\). For part (c), \( (-7n)^{-1}\) follows similarly, giving \(\frac{1}{-7n}\) which is also \(-\frac{1}{7n}\). Mastering these rules can make working with more complex expressions straightforward.

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