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Chapter 4: Problem 5

True or False? \(x^{-1}=\frac{1}{x}\)

Short Answer

Expert verified
True

Step by step solution

01

Understand The Meaning of the Negative Exponent

In mathematics, a negative exponent indicates the reciprocal of the base. So, \(x^{-1}\) then means the reciprocal of x. That is one divided by the base (x), which can be written as \(\frac{1}{x}\).
02

Comparison

Having understood the meaning of \(x^{-1}\), it can be seen that it is equal to \(\frac{1}{x}\), which is just another way of expressing the reciprocal of x. Therefore, the initial statement is correct.

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Key Concepts

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

Reciprocal
When dealing with numbers, a reciprocal is the number you multiply with a given number to result in one. Essentially, for any number or expression like 'x', the reciprocal is represented by \( \frac{1}{x} \). It's often used to simplify expressions or solve equations.

  • For example, the reciprocal of 5 is \( \frac{1}{5} \).
  • Similarly, the reciprocal of \( \frac{1}{3} \) is 3, because \( \frac{1}{3} \times 3 = 1 \).
Understanding the idea of reciprocals helps in various areas of mathematics, especially when working with fractions or solving equations. So, in the context of exponents, recognizing that a negative exponent introduces a reciprocal is crucial.
Exponents
Exponents are a shorthand notation used in mathematics to express how many times a number, the base, is multiplied by itself. For instance, \( x^3 \) means \( x \times x \times x \).
  • Positive exponents indicate multiplication.
  • Negative exponents, like \( x^{-1} \), suggest taking the reciprocal.
  • An exponent of zero, such as \( x^0 \), always equals 1.
Negative exponents simplify expressions, especially when dividing numbers or dealing with fractional expressions. For instance, when we encounter \( x^{-1} \), it equates to \( \frac{1}{x} \), due to its inherent reciprocal property.
Mathematics
Mathematics is a field that explores numbers, equations, structures, and their interrelationships. Within this vast domain, exponents play a critical role in expressing large numbers or simplifying expressions.

In particular:
  • They allow writing repeated multiplication succinctly.
  • Negative exponents enable transforming division into multiplication using reciprocals.
  • Understanding these concepts is vital for solving equations effectively.
The fundamental relationship between negative exponents and reciprocals, as exemplified by \( x^{-1} = \frac{1}{x} \), showcases the beauty and efficiency of mathematical notation.

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Most popular questions from this chapter

The function \(f(x)=|x+2|\) is not one-to-one. How can the domain of \(f\) be restricted to produce a one-to-one function?

Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{2} 12$$

Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{3} 1.25$$

Solve the logarithmic equation and eliminate any extraneous solutions. If there are no solutions, so state. $$\log (x+5)-\log \left(4 x^{2}+5\right)=0$$

Use the change-of-base formula to evaluate each logarithm using a calculator. Round answers to four decimal places. $$\log _{3} 2.75$$
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