Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 3: Problem 48
Evaluate the expression and write the result in the form \(a+b i.\) $$\sqrt{\frac{-9}{4}}$$
Short Answer
Expert verified
The result is \(0 + \frac{3}{2}i\).
Step by step solution
01
Understand the Problem
We need to evaluate the expression \( \sqrt{\frac{-9}{4}} \) and express the result in the form \( a + b i \), which is the standard form for complex numbers.
02
Simplify the Square Root
The expression has a negative number under the square root, so we need to factor out \(-1\). Rewriting the expression: \( \sqrt{\frac{-9}{4}} = \sqrt{\frac{-9}{4}} = \sqrt{\frac{-1 \cdot 9}{4}} \).
03
Separate the Components
We know that \( \sqrt{\frac{-1 \cdot 9}{4}} \) can be separated into \( \frac{\sqrt{-1} \cdot \sqrt{9}}{\sqrt{4}} \).
04
Evaluate Each Component
Evaluate each square root separately. We have \( \sqrt{-1} = i \) (where \( i \) is the imaginary unit), \( \sqrt{9} = 3 \), and \( \sqrt{4} = 2 \).
05
Combine the Results
Substitute back into the expression to get \( \frac{i \cdot 3}{2} = \frac{3i}{2} \).
06
Express in Standard Form
The expression \( \frac{3i}{2} \) is already in the form \( a + bi \) where \( a = 0 \) and \( b = \frac{3}{2} \).
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
In mathematics, the imaginary unit is a fundamental concept that often puzzles beginners. The imaginary unit is denoted as \( i \) and is defined as the square root of -1. This can be confusing since square roots usually result in a positive or negative real number. However, for negative numbers under the square root, \( i \) provides a new way to define such results.
- Definition: \( i = \sqrt{-1} \)
- Key property: \( i^2 = -1 \)
Square Roots
Square roots are a basic mathematical concept that involves finding a number which, when multiplied by itself, gives the original number. It's expressed by the radical symbol \( \sqrt{} \). When dealing with square roots of negative numbers, like in our exercise where we have \( \sqrt{\frac{-9}{4}} \), things get interesting!
- For positive numbers, the square root gives a real number.
- For negative numbers, we use the imaginary unit \( i \) and proceed as \( \sqrt{-x} = \sqrt{x} \times i \).
Expression Simplification
Expression simplification involves reducing an equation or expression to its simplest form. For the expression \( \sqrt{\frac{-9}{4}} \), we aim to express it in the standard complex number form \( a + bi \).
Thus, the solution is \( a = 0 \) and \( b = \frac{3}{2} \), neatly fitting the complex form \( a + bi \). Simplifying expressions brings clarity, ensuring results are in a universally recognized form.
- Factoring out negative components: Convert negative square roots using \( i \).
- Breaking down components: Treat each term in the fraction individually, separating numerators and denominators.
- Combining terms: Take the simplified parts and recombine them following the properties of fractions and complex numbers.
Thus, the solution is \( a = 0 \) and \( b = \frac{3}{2} \), neatly fitting the complex form \( a + bi \). Simplifying expressions brings clarity, ensuring results are in a universally recognized form.
