Problem 25 - A Graphical Approach to College Algebra

Question

Write each number in simplest form, without a negative radicand. $$-\sqrt{-39}$$

Step-by-Step Solution

Answer

The simplest form is \(-i\sqrt{39}\).

Step 1: Review the Expression

The expression given is \(-\sqrt{-39}\). We need to simplify this, ensuring no negative radicand.

Step 2: Handle the Negative Inside the Square Root

The negative inside the square root \(-39\) can be rewritten using the imaginary unit \(i\), where \(i = \sqrt{-1}\). This gives us \(\sqrt{-39} = \sqrt{39} \cdot \sqrt{-1} = \sqrt{39} \cdot i\).

Step 3: Include the Negative Sign from Outside

The original expression includes a negative sign outside the square root, so it becomes \(-\sqrt{39} \cdot i = -i\sqrt{39}\).

Step 4: Simplify the Expression

Since \(39\) cannot be simplified further into a product of simpler square numbers, the expression \(-i\sqrt{39}\) is in its simplest form.

Key Concepts

Imaginary Unit

The imaginary unit is a cornerstone concept in complex numbers. This mysterious number, represented as \(i\), is defined by the equation \(i = \sqrt{-1}\). It is a mathematical invention that allows us to work with the square root of negative numbers. Normally, you cannot take the square root of a negative number within the real number system. However, by introducing this new concept \(i\), we extend our number system to include these otherwise impossible quantities.

• \(i^2 = -1\): This relationship is foundational in handling complex numbers.
• Any real number multiplied by \(i\) becomes imaginary.
• Combining real and imaginary numbers results in complex numbers.

Understanding \(i\) opens up a new mathematical world. Complex numbers are used in a variety of fields, such as engineering, physics, and even art.

Simplifying Radicals

Simplifying radicals involves expressing a radical term with the simplest possible terms. A key step often includes factoring the number inside the radical into its prime factors. Once we've done this, we can simplify the expression by "bringing out" any pairs of numbers from under the square root symbol.For example, simplifying \(\sqrt{48}\) involves recognizing that \(48 = 2 \times 2 \times 2 \times 2 \times 3\). We can pair two 2's together to take them out of the square root:- \(\sqrt{48} = \sqrt{2^2 \times 2^2 \times 3} = 2 \times 2 \times \sqrt{3} = 4\sqrt{3}\)This process of simplification relies on extracting these pairs, which represents complete square numbers. The remaining numbers under the radical remain as they are, which is what happened in the solution where \(\sqrt{39}\) was left unchanged since 39 only factors into 3 and 13, which do not form a pair.

Negative Radicand

A negative radicand is simply a negative number within a square root, like \(\sqrt{-39}\). In standard arithmetic with real numbers, this would be considered undefined. However, with the introduction of imaginary numbers, we can make sense of it. By representing the negative sign inside the square root with the imaginary unit \(i\), we rewrite the expression:- For example, \(\sqrt{-39}\) is expressed as \(\sqrt{39} \times \sqrt{-1} = \sqrt{39} \cdot i\).This technique allows us to always keep the radicand (the number inside the square root) non-negative, simplifying the handling of such numbers in calculations. This turns potentially confusing, negative radicands into manageable expressions using the imaginary unit. By rearranging the terms, we maintain mathematical consistency and open possibilities for more advanced calculations in complex mathematics.