Problem 4 - Algebra 2

Question

Simplify each number by using the imaginary number \(i\) $$ \sqrt{-81} $$

Step-by-Step Solution

Answer

The simplified form of \(\sqrt{-81}\) is \(9i\).

Step 1: Identify the negative under the square root

Notice that the square root of a negative number is not real. It can be expressed in terms of the imaginary unit, which is defined as \(i = \sqrt{-1}\).

Step 2: Express the square root of -81 in terms of i

Write \(\sqrt{-81}\) as \(\sqrt{-1}\cdot\sqrt{81}\). This allows you to separate the negative under the square root into the imaginary unit.

Step 3: Simplify the expression

Now, simplify the two separated parts. We know that \(i = \sqrt{-1}\) and \(\sqrt{81} = 9\). So the expression becomes \(i \cdot 9\), which is simplified to \(9i\).

Key Concepts

Simplifying Square Roots

Understanding the concept of square roots is essential for working with complex numbers and the imaginary unit. A square root, symbolized by \( \sqrt{\cdot} \) is a value that, when multiplied by itself, returns the original number. Simplifying square roots involves finding this value. However, a new challenge arises when we encounter negative numbers under the square root, as they aren't part of the real numbers we work with on a daily basis. To resolve this, we introduce the concept of imaginary numbers.

Let's say you need to simplify \( \sqrt{-81} \). Since the root of a negative number doesn't exist within the set of real numbers, we first identify the square root of a positive number and then attribute the negative aspect to the imaginary unit. We break it down into \( \sqrt{-1} \) and \( \sqrt{81} \), simplifying the latter to 9 since \( 9 \times 9 = 81 \). To deal with \( \sqrt{-1} \), we use the imaginary unit 'i'.

Imaginary Unit

The imaginary unit, denoted as 'i', is a fundamental concept when dealing with the square roots of negative numbers. It is defined by the relation \( i = \sqrt{-1} \). This unit lays the groundwork for complex numbers and expands our number system beyond the real numbers. It's important to note that the imaginary unit isn't a figure of speech; it's a practical and critical component of modern mathematics, used in fields such as electrical engineering and quantum physics.

When simplifying square roots like \( \sqrt{-81} \), the role of the imaginary unit becomes clear. By expressing \( \sqrt{-81} \) as \( i \cdot \sqrt{81} \), we are effectively using the properties of 'i' to navigate around the impossibility of finding real square roots of negative numbers. In doing so, we ensure every number, even a negative one, has a square root within the extended complex number system.

Complex Numbers

Complex numbers are an extension of the real numbers and are composed of two parts: a real part and an imaginary part. A complex number is typically written in the form \( a + bi \), where 'a' is the real part, 'b' is the imaginary part, and 'i' represents the square root of -1. Complex numbers can be added, subtracted, multiplied, and divided, much like real numbers, but they require some additional rules to handle the imaginary unit.

To see complex numbers in action, consider the example of simplifying \( \sqrt{-81} \). The solution involves the imaginary unit: \( 9i \), which is a simple form of a complex number where the real part is zero (0 + 9i). Complex numbers include the entire set of real numbers, as any real number 'r' can be expressed as \( r + 0i \), showing the fluid connection between these two sets. Grasping the concept of complex numbers is crucial not only in theoretical mathematics but also in practical applications like signal processing, which rely heavily on this number system.