Problem 30 - College Algebra Essentials

Question

In Exercises \(29-44\), perform the indicated operations and write the result in standard form. $$ \sqrt{-81}-\sqrt{-144} $$

Step-by-Step Solution

Answer

The result of \( \sqrt{-81}-\sqrt{-144} \) is \( -3i \).

Step 1: Understand the Concept of a Complex Number

A complex number is made up of a real part and an imaginary part. It is important to understand that square root of -1 is represented by 'i', which is an imaginary number. Therefore, the square root of any negative number can be represented as the square root of the corresponding positive number, multiplied by 'i'.

Step 2: Evaluate the Square Roots

Firstly, let's break down the negative square roots into their complex representations. The square root of -81 can be written as \( \sqrt{81} \times \sqrt{-1} \), which simplifies to \( 9i \). Similarly, the square root of -144 can be written as \( \sqrt{144} \times \sqrt{-1} \), which simplifies to \( 12i \).

Step 3: Perform the Subtraction

As proposed in the initial operation, let's subtract these two numbers. That means \( 9i - 12i \). When you subtract these, you get \( -3i \).

Key Concepts

Imaginary Numbers

When venturing beyond the realm of real numbers, we encounter imaginary numbers, which have fascinated mathematicians for centuries. At their heart is the unit imaginary number, represented by the symbol 'i'. It's defined as the square root of -1, which doesn't have a solution within the set of real numbers. Imaginary numbers allow us to perform operations involving square roots of negative numbers.

For example, the square root of -81 is not possible in the set of real numbers, but in terms of imaginary numbers, it's written as \( \text{i} \times \text{i} \times 9 = 9i \). This concept also applies to other negative square roots, turning the impossible into the possible within the expanded universe of complex numbers.

Operations with Complex Numbers

Working with complex numbers involves operations such as addition, subtraction, multiplication, and division much like with real numbers. However, special rules apply when dealing with the imaginary part. For instance, to add or subtract complex numbers, we combine their real parts and imaginary parts separately.

When we come across a subtraction like \( 9i - 12i \), we treat 'i' as a common factor and simply subtract the coefficients, leading to \( -3i \). In multiplication, we use the distributive property and the fact that \( i^2 = -1 \) to simplify the expressions. Division typically requires a process called 'complex conjugation' to eliminate imaginary numbers from the denominator.

Standard Form of Complex Numbers

The standard form is the most common way to express a complex number, and it is written as \( a + bi \), where 'a' is the real part and 'b' is the coefficient of the imaginary part. When we're given expressions involving \textbackslash sqrt{-81}-\textbackslash sqrt{-144}\textbackslash, our job is to simplify them into the standard form.

In the solution provided, we began with this step and simplified the square roots to obtain imaginary numbers. The final answer, \( -3i \), is already in standard form because there's no real part — the real part is effectively zero. Thus, \( -3i \) can be written as \( 0 - 3i \), adhering to the standard form convention.