Problem 17 - Algebra 2 and Trigonometry

Question

In \(3-18,\) write each number in terms of \(i\) $$ -4-\sqrt{-24} $$

Step-by-Step Solution

Answer

The expression in terms of \(i\) is \(-4 - 2\sqrt{6}i\).

Step 1: Identify the Imaginary Part

The first part of solving this is to look at the term \(\sqrt{-24}\). Notice that inside the square root, the negative sign indicates an imaginary number. So, \(\sqrt{-24} = \sqrt{24} \times \sqrt{-1}\) which simplifies to \(\sqrt{24} \times i\).

Step 2: Simplify the Radical

Further simplify \(\sqrt{24}\). Break it down into factors: \(\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6}\). This gives us \(2\sqrt{6}\). Now substitute back to obtain \(\sqrt{-24} = 2\sqrt{6} i\).

Step 3: Express the Full Expression in Terms of \(i\)

Combine the real and imaginary parts in the expression \(-4 - \sqrt{-24}\). Substituting the simplified form, we have: \(-4 - 2\sqrt{6} i\).

Key Concepts

Imaginary Numbers

Complex numbers consist of a real part and an imaginary part. An imaginary number typically includes the unit imaginary number, denoted using the symbol \(i\), where \(i\) is defined as the square root of \(-1\). This definition stems from the fact that no real number squared gives a negative result. So, \(i^2 = -1\). Imaginary numbers help us solve equations that do not have solutions in the set of real numbers.

In the context of the expression \(-4-\sqrt{-24}\), the term \(\sqrt{-24}\) involves an imaginary part because of the square root of a negative number. This concept helps us to understand how to work with non-real solutions and extend our number system to include not only real numbers but also imaginary and complex numbers.

Simplifying Radicals

Simplifying radicals involves breaking down a radical expression into its simplest form. When you encounter a radical with a negative number under the square root, as in \(\sqrt{-24}\), you can start by expressing it in terms of \(i\).

To simplify \(\sqrt{-24}\), you multiply it with \(\sqrt{-1}\) to separate the real and imaginary components: \(\sqrt{-24} = \sqrt{24} \times \sqrt{-1}\). The negative inside the square root signifies that it is an imaginary number, which translates into \(i\).

Next, break down \(\sqrt{24}\) into its prime factors: \(\sqrt{4\times6}\). Simplify further to get \(\sqrt{4} \times \sqrt{6}\), which results in \(2\sqrt{6}\). So, \(\sqrt{-24} = 2\sqrt{6}i\). Radicals are often simplified this way to make equations easier to work with.

Expressions in Terms of i

After recognizing and simplifying \(\sqrt{-24}\) to \(2\sqrt{6}i\), it’s essential to express the entire original expression in terms of \(i\). This involves combining the real and imaginary parts of the expression. Here, the real number is \(-4\), and the imaginary component is \(-2\sqrt{6}i\).

Writing complex numbers in the form \(a + bi\) is useful because it reinforces the structure of complex numbers. In this exercise, we write \(-4 - \sqrt{-24}\) ultimately as \(-4 - 2\sqrt{6}i\).

This expression provides a clear separation between the real and imaginary parts, enabling further mathematical operations like addition, subtraction, and multiplication within the realm of complex numbers. Understanding how to rewrite expressions in terms of \(i\) provides a solid foundation for tackling more advanced problems involving complex numbers.