/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 19 Find the reciprocal of each numb... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 19

Find the reciprocal of each number. $$3, \quad-\frac{4}{\sqrt{3}}, \quad \frac{y}{b}$$

Short Answer

Expert verified
Reciprocals are \(\frac{1}{3}\), \(-\frac{\sqrt{3}}{4}\), and \(\frac{b}{y}\).

Step by step solution

01

Understanding Reciprocals

The reciprocal of a number is defined as a number that, when multiplied with the original number, yields 1. Mathematically, if a number is represented as \(x\), its reciprocal is \(\frac{1}{x}\). We will apply this concept to find the reciprocal of each given number.
02

Reciprocal of 3

The number is 3. Its reciprocal is obtained by the formula \(\frac{1}{x}\). Thus, the reciprocal of 3 is \(\frac{1}{3}\).
03

Reciprocal of \(-\frac{4}{\sqrt{3}}\)

For the fraction \(-\frac{4}{\sqrt{3}}\), its reciprocal is found by flipping the numerator and denominator, resulting in \(-\frac{\sqrt{3}}{4}\).
04

Reciprocal of \(\frac{y}{b}\)

For the fraction \(\frac{y}{b}\), the reciprocal is obtained by inverting it. Thus, the reciprocal is \(\frac{b}{y}\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Reciprocal
The reciprocal of a number is like its mirror image when it comes to multiplication. When we multiply a number by its reciprocal, the result is always 1. For any number represented as \( x \), the reciprocal is written as \( \frac{1}{x} \). This means we flip the number upside down, especially noticeable with fractions. For simple numbers, like 3, the reciprocal is straightforward: \( \frac{1}{3} \).
  • For a number \( a/b \), the reciprocal is \( b/a \).
  • Working with reciprocals is crucial in dividing fractions because dividing is the same as multiplying by the reciprocal.

Understanding this concept helps in solving many arithmetic problems easily and shows the graceful symmetry in math. So next time you see the number 3, remember its flip side is \( \frac{1}{3} \)!
Fractions
Fractions are the representation of numbers in the form of \( a/b \), where \( a \) is the numerator (the top part) and \( b \) is the denominator (the bottom part). These numbers help express parts of a whole, making them very handy in mathematics.
  • Mixed fractions have a whole number and a fraction, like \( 2\frac{1}{2} \).
  • Remember that the reciprocal of \( a/b \) is \( b/a \), swapping the places of the numerator and the denominator.

Fractions can sometimes look confusing, especially with square roots or variables, but the rules remain the same. For example, the reciprocal of \(-\frac{4}{\sqrt{3}}\) involves flipping, resulting in \(\frac{-\sqrt{3}}{4}\). Understanding these basic rules makes tackling problems with fractions far easier.
Arithmetic
Arithmetic is all about numbers and the basic operations we perform on them, such as addition, subtraction, multiplication, and division. When dealing with reciprocals and fractions, it involves quite a bit of multiplication and division.
  • Multiplying numbers means scaling them up, while dividing involves reciprocals.
  • With fractions, multiplication is done by multiplying the numerators and denominators across, and division is managed by converting the divisor to its reciprocal.

For example, to divide by a fraction like \(\frac{y}{b}\), you multiply by its reciprocal \(\frac{b}{y}\). This transforms complex operations into straightforward multiplication, making arithmetic with fractions much simpler. By mastering such techniques, you gain a critical tool for almost every math problem.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Make the indicated coversions. $$65.2 \frac{\mathrm{m}}{\mathrm{s}} \text { to } \frac{\mathrm{ft}}{\min }$$

Perform the indicated operations assuming all numbers are approximate. Round your answers using the procedure shown in Example 11 $$\frac{8.97(4.003)}{2.0+4.78}$$

Assume that all numbers are approximate unless stated otherwise. The percent of alcohol in a certain car engine coolant is found by performing the calculation \(\frac{100(40.63+52.96)}{105.30+52.96} .\) Find this percent of alcohol. The number 100 is exact.

For computer memory, the metric prefixes have an unusual meaning: \(1 \mathrm{KB}=2^{10}\) bytes, \(1 \mathrm{MB}=2^{10} \mathrm{KB}, 1 \mathrm{GB}=2^{10} \mathrm{MB},\) and I \(\mathrm{TB}=2^{10}\) GB. How many bytes are there in 1 TB? (KB is kilobyte, MB is megabyte, GB is gigabyte, TB is terabyte)

Perform the indicated multiplications. By multiplication, show that \((x+y)\left(x^{2}-x y+y^{2}\right)=x^{3}+y^{3}\).
See all solutions

Recommended explanations on Math Textbooks

Geometry

Read Explanation

Applied Mathematics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.