Problem 48 Evaluate the radical expression ... [FREE SOLUTION]

91影视

Precalculus Mathematics for Calculus

James Stewart, Lothar Redlin, Saleem Watson

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 3: Problem 48

Evaluate the radical expression and express the result in the form $a+b i$ $$\sqrt{\frac{-9}{4}}$$

Short Answer

Expert verified

The expression $ \sqrt{\frac{-9}{4}} $ in the form $ a + bi $ is $ 0 + \frac{3}{2}i $.

Step by step solution

Separate the Radical into Real and Imaginary Parts

Start by re-writing the original expression $ \sqrt{\frac{-9}{4}} $ as $ \sqrt{-1} \times \sqrt{\frac{9}{4}} $. This is possible because of the property $ \sqrt{a \times b} = \sqrt{a} \times \sqrt{b} $ when dealing with non-negative numbers \, \text{(except when dealing with complex numbers)}.

Evaluate $ \sqrt{-1} $

Recognize that $ \sqrt{-1} = i $, where $ i $ is the imaginary unit. The imaginary unit $ i $ satisfies $ i^2 = -1 $.

Evaluate $ \sqrt{\frac{9}{4}} $

Find the square root of $ \frac{9}{4} $ by taking the square root of the numerator and the denominator separately, which gives us $ \frac{\sqrt{9}}{\sqrt{4}} = \frac{3}{2} $.

Combine the Results

Multiply the results from Step 2 and Step 3 together: $ i \times \frac{3}{2} = \frac{3}{2}i $.

Express in Form $ a + bi $

Write the final answer in the standard form of a complex number. In this expression, there is no real part, so it is $ 0 + \frac{3}{2}i $.

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.

Imaginary Unit

The imaginary unit is denoted by the letter $i$. It is a fundamental concept in the field of complex numbers. The essence of $i$ is to provide a solution for the equation $x^2 = -1$, which has no solutions among real numbers. Thus, $i$ is defined such that $i^2 = -1$. This concept introduces the idea of numbers that involve the square roots of negative numbers. Whenever you encounter the square root of a negative number, $i$ becomes significant.

$i$ squared equals -1: $i^2 = -1$
$i$ is used to express imaginary parts of complex numbers

Understanding the imaginary unit is crucial for dealing with complex numbers and expressions that result in the square roots of negative values.

Radical Expression

A radical expression involves a root symbol, often indicating a square root or higher roots such as cube roots. In this context, the expression $\sqrt{\frac{-9}{4}}$ is a radical expression. It represents the square root of the fraction $\frac{-9}{4}$, which includes a negative sign inside the radical.To simplify a radical expression, you often separate it into real and imaginary components, especially when dealing with a negative number under the square root. This can be done using properties of radicals that allow the separation of multiplication within the root.

Radical expressions can involve real numbers, complex numbers, or both
Breaking down radicals helps to isolate imaginary units

The careful handling of radicals with negative values is a stepping stone to understanding complex numbers.

Square Root

The square root of a number $x$ is a value that, when multiplied by itself, yields $x$. The square root is denoted as $\sqrt{x}$.In the context of complex numbers, when you encounter a negative under the square root, the result transforms into an imaginary number. $\sqrt{-1}$ specifically transforms into the imaginary unit $i$.

The square root is fundamental when dealing with both real and complex numbers
For positive numbers, square roots have two values: a positive and a negative

Understanding square roots is integral for simplifying expressions and handling both real and imaginary components.

Standard Form of a Complex Number

The standard form of a complex number is expressed as $a + bi$, where $a$ and $b$ are real numbers. Here, $a$ represents the real part, and $bi$ represents the imaginary part of the complex number.The complex number is a combination of both a real and an imaginary part. This form is very useful for mathematical calculations and solving equations involving complex numbers.

The real part ($a$) can be zero, making the entire number purely imaginary, as in $0 + bi$
Both parts of a complex number are crucial for its properties

Using the standard form makes complex numbers easier to visualize and manipulate in equations.

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Evaluate the radical expression and express the result in the form \(a+b i\) $$\sqrt{\frac{-9}{4}}$$

Short Answer

Step by step solution

Separate the Radical into Real and Imaginary Parts

Evaluate \( \sqrt{-1} \)

Evaluate \( \sqrt{\frac{9}{4}} \)

Combine the Results

Express in Form \( a + bi \)

Key Concepts

Imaginary Unit

Radical Expression

Square Root

Standard Form of a Complex Number

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Theoretical and Mathematical Physics

Pure Maths

Discrete Mathematics

Geometry

Probability and Statistics

Study anywhere. Anytime. Across all devices.