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Chapter 1: Problem 9

Give the reciprocal of each number. \(-\frac{1}{\sqrt{2}}\)

Short Answer

Expert verified
The reciprocal is \(-\sqrt{2}\).

Step by step solution

01

Understand the Concept of Reciprocal

The reciprocal of a number is simply 1 divided by that number. If the number is a fraction, the reciprocal is obtained by flipping the numerator and the denominator. For any non-zero number \( a \), the reciprocal is \( \frac{1}{a} \).
02

Identify the Fraction Components

The given number is \( -\frac{1}{\sqrt{2}} \), which has a numerator of \(-1\) and a denominator of \(\sqrt{2}\).
03

Flip the Fraction

To find the reciprocal, flip the numerator and the denominator of the fraction. This results in \(-\sqrt{2}\).
04

Simplify the Answer

Although \(-\sqrt{2}\) is already in a simplified form, if we needed a more rationalized form, it would still result in \(-\sqrt{2}\) because it already complies with standard simplification.

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Key Concepts

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

Understanding Fractions
In mathematics, fractions are a way of expressing numbers that are not whole. A fraction consists of two main parts: a numerator and a denominator. Fractions are an essential concept because they allow us to represent values that lie between whole numbers in a precise manner.
  • Fractions are expressed as \ \( \frac{a}{b} \ \), where \ \( a \ \) is the numerator and \ \( b \ \) is the denominator.
  • Fractions can represent parts of a whole, portions of sets, or even results of division operations.
  • Understanding fractions is crucial for performing various mathematical operations, such as addition, subtraction, multiplication, and division of these numbers.
By mastering fractions, you can better handle scenarios that involve measurements and calculations in daily life.
Defining the Numerator
The numerator is the top part of a fraction. It tells you how many parts of the whole you have. For instance, in the fraction \ \( \frac{3}{5} \ \), the numerator is 3.
  • The numerator indicates the number of parts being considered from a division or subset of the denominator.
  • When working with fractions, the numerator provides clarity by specifying how many parts are being discussed or represented.
  • The numerator is crucial in operations such as flipping a fraction to find its reciprocal.
Understanding the numerator's role helps to perform operations accurately and understand the represented value more clearly.
What is a Denominator?
The denominator is the bottom part of a fraction. It tells you into how many equal parts the whole is divided. If you look at a fraction like \ \( \frac{3}{5} \ \), the denominator is 5.
  • The denominator determines the scale or size of each part of the whole.
  • Knowing the denominator is essential for comparing or converting fractions, as it allows for clear understanding of how the parts relate to the total.
  • In the process of finding reciprocals, the denominator swaps with the numerator.
A firm grasp of how denominators function in fractions aids in solving problems that involve sharing or distributing resources evenly.
Exploring the Square Root
A square root is a value that, when multiplied by itself, gives the original number. For example, the square root of 9 is 3 because \ \(3 \times 3 = 9\ \). Square roots are often used in simplifying expressions and solving quadratic equations.
  • Square roots show up frequently in various areas of mathematics, including geometry and algebra.
  • You can express the square root of a number \ \( x \ \) using the symbol \ \( \sqrt{x} \ \).
  • Understanding square roots is crucial when dealing with operations that require rationalizing denominators in fractions that involve roots.
Mastering the concept of square roots enables you to handle calculations that involve powers and roots accurately and confidently.

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Most popular questions from this chapter

Use a ratio identity to find \(\cot \theta\) given the following values. \(\sin \theta=-\frac{5}{13}\) and \(\cos \theta=-\frac{12}{13}\)

Use the equivalent forms of the first Pythagorean identity on Problems 31 through 38 . Find \(\csc \theta\) if \(\cot \theta=\frac{24}{7}\) and \(\theta\) terminates in QIII.

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If \(\cos \theta=\frac{1}{4}\), find \(\cos ^{2} \theta\). a. \(\frac{1}{16}\) b. \(\cos ^{2} \frac{1}{16}\) c. \(\frac{1}{2}\) d. 16
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