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Chapter 1: Problem 22

Write each expression in terms of \(i\) and simplify. (See Examples \(1-2)\) $$\sqrt{-23}$$

Short Answer

Expert verified
\sqrt{-23} = i \sqrt{23}.

Step by step solution

01

Recognize the Negative Sign Inside the Square Root

Identify that there is a negative sign inside the square root. The expression can be written as \ \ \ \ \ \ \ \ \ \ \ \ \ \( \sqrt{-23} \).
02

Use the Property of Square Roots with Negative Numbers

Recall that \sqrt{-1} = i \. Apply this property to the expression so that \ \ \ \ \ \ \ \ \ \ \ \ \ \( \sqrt{-23} = \sqrt{23} \cdot \sqrt{-1} \ = \sqrt{23} \cdot i \).
03

Simplify the Expression

Combine the constants and \i \ into one expression. Thus, the expression simplifies to \ \ \ \ \ \ \ \ \ \ \ \ \ \( \sqrt{-23} = i \sqrt{23} \).

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Key Concepts

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

imaginary unit
Understanding the imaginary unit, represented as \( i \), is crucial when working with complex numbers. The imaginary unit is defined by the property \( i^2 = -1 \). This property allows us to handle the square roots of negative numbers, which are not defined within the set of real numbers.
For example:
  • When you see \( \sqrt{-1} = i \), it means that you are working with an imaginary number.
  • Building on this, \( i \) is used as a tool to simplify expressions involving the square roots of negative numbers.
Understanding this fundamental concept makes it easier to work with complex numbers, and it is the foundation for many operations involving them.
square root of negative numbers
Taking the square root of negative numbers involves incorporating the imaginary unit \( i \). Here鈥檚 a simple way to think about it:
When you encounter a square root of a negative number, say \( \sqrt{-a} \), you can break it down as follows:
  • First, identify the negative sign: \( -a \) means there is a negative number inside the square root.
  • Use the property \( \sqrt{-1} = i \) to separate the negative sign.
  • Express it as: \ \sqrt{-a} = \sqrt{a} \cdot \sqrt{-1} = \sqrt{a} \cdot i \.
This method allows you to convert any square root of a negative number into a form involving \( i \), making it easier to handle in mathematical operations and simplifications.
simplification
Simplifying expressions that involve complex numbers and the imaginary unit \( i \) follows a few straightforward steps:
Let鈥檚 take an example from the exercise:
  • Start with the expression: \( \sqrt{-23} \).
  • Identify the negative sign and recognize you need to use \( i \).
  • Re-write the expression as: \ \sqrt{-23} = \sqrt{23} \cdot \sqrt{-1} = \sqrt{23} \cdot i \.
  • Combine the constants and \( i \) into one simplified expression: \ \sqrt{-23} = i \sqrt{23} \.
By following these steps:
1. Identify negative signs inside square roots.
2. Apply the property \( \sqrt{-1} = i \).
3. Combine and simplify the resulting expression.
These steps help make complex numbers easier to understand and work with in various mathematical contexts.

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