Problem 30 - A Graphical Approach to College Algebra

Question

Write each number in simplest form, without a negative radicand. $$-11-\sqrt{-24}$$

Step-by-Step Solution

Answer

\(-11 - 2\sqrt{6}i\)

Step 1: Identify the Complex Number

The expression \(-11-\sqrt{-24}\) is not currently in simplest form because it has a negative radicand within the square root. This means it involves complex numbers, as the square root of a negative number is not real.

Step 2: Express the Imaginary Part

Recognize that \(\sqrt{-24}\) can be rewritten as \(\sqrt{24} \cdot i\), since \(i\) is the imaginary unit where \(i = \sqrt{-1}\).

Step 3: Simplify the Radicand

Factor 24 inside the square root to simplify: \(\sqrt{24} = \sqrt{4 \times 6}\). This can be simplified to \(2\sqrt{6}\) since the square root of 4 is 2.

Step 4: Combine Parts

Replace \(\sqrt{-24}\) with \(2\sqrt{6}i\). Thus, the expression \(-11-\sqrt{-24}\) simplifies to \(-11 - 2\sqrt{6}i\).

Step 5: Write the Simplified Form

The simplest form of the number \(-11-\sqrt{-24}\) without a negative radicand under the square root is \(-11 - 2\sqrt{6}i\).

Key Concepts

Simplifying Radicals

Radicals are expressions that involve square roots or other root operations. To simplify radicals, the goal is to break down the expression under the radical sign to its simplest form. This typically involves factoring the number under the radical into its prime components.

• Identify the factors of the number under the radical.
• Look for square numbers (like 4, 9, 16, etc.) which can be easy to take the square root of.
• If there is a square number factor, take the square root of it outside the radical.

For example, in the expression \(\sqrt{24}\), you can write 24 as \(4 \times 6\). Since 4 is a perfect square, its square root is 2, which can be taken outside the radical. This simplifies \(\sqrt{24}\) to \(2\sqrt{6}\). Repeated practice with similar expressions will make simplifying radicals second nature.

Imaginary Unit

In mathematics, the imaginary unit is denoted by \(i\) and is defined as the square root of -1. This concept is crucial when dealing with square roots of negative numbers, because it allows such numbers to be expressed in a meaningful way.

• The imaginary unit \(i\) is used when the radicand (the number under the root) is negative.
• It helps to convert square roots of negative numbers into a form that mathematicians can work with.

For example, \(\sqrt{-1} = i\). Therefore, any time you encounter a square root of a negative number, like \(\sqrt{-24}\), you use the imaginary unit to rewrite it as \(\sqrt{24} \cdot i\). This is a foundational concept in complex numbers, making solutions possible where they otherwise would not be.

Negative Radicand

A radicand is the number or expression inside a radical symbol. When dealing with square roots, a negative radicand presents a unique situation, as the square root of a negative number is not defined among the set of real numbers. This is where complex numbers and the imaginary unit come into play.

• When a radicand is negative, the expression involves imaginary numbers.
• The negative sign under a square root turns the expression into a complex number by introducing \(i\).

In our given example, \(-\sqrt{-24}\), the negative radicand \(-24\) is rewritten using \(i\), leading to the expression \(\sqrt{24} \cdot i\). It shows that any time you face a negative radicand, you are dealing with possibilities beyond real numbers, exploring the more expansive field of complex numbers. Through this exercise, the concept of a negative radicand becomes clear and manageable in the context of complex numbers.