Problem 25 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

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

Step 1: Simplify Negative Under Square Root

We start with the expression \(-\sqrt{-39}\). Since we cannot have a negative number under a square root in real numbers, we'll use the imaginary unit \(i\), where \(i = \sqrt{-1}\). This means \(\sqrt{-39} = \sqrt{-1 \times 39} = \sqrt{-1} \times \sqrt{39} = i\sqrt{39}\).

Step 2: Address the Original Expression

Now replace the expression \(-\sqrt{-39}\) with our simplification: \(-i\sqrt{39}\). This expression is the simplest form, as it removes the negative radicand by expressing it using the imaginary unit \(i\).

Key Concepts

Complex Numbers

Complex numbers are numbers that include both real and imaginary parts. They are written in the form of \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The imaginary unit \(i\) is defined as \(\sqrt{-1}\), which allows us to handle square roots of negative numbers.

For example, in the expression \(-i\sqrt{39}\), the component \(-i\) represents the imaginary unit, and \(\sqrt{39}\) is a real number. This combination forms a complex number, as it has an imaginary part. Complex numbers extend our number system, enabling us to solve equations that have no solutions within the real numbers alone, such as \(x^2 + 1 = 0\). By introducing \(i\), we make operations with square roots of negative numbers possible, adding a new dimension to our calculations.

Square Roots

Square roots are a fundamental operation in mathematics, representing a value that, when multiplied by itself, gives the original number. In the case of \(\sqrt{39}\), it means finding the number that squared results in 39. Square roots are straightforward for positive numbers.

However, when dealing with negative numbers under the square root, such as \(\sqrt{-39}\), we need to incorporate imaginary numbers, using \(i\) where \(i = \sqrt{-1}\). This helps us "extract" the negative, allowing us to simplify expressions involving square roots of negative values. For instance, \(\sqrt{-39} = i \sqrt{39}\), representing both the operation and the involvement of imaginary units to solve the problem.

Simplification of Expressions

Simplifying mathematical expressions involves reducing them to their simplest form without losing their value. It's essential for clarity and ease of manipulation. When simplifying expressions that include square roots of negative numbers, like \(-\sqrt{-39}\), using imaginary units can help us achieve simpler forms.

We start by recognizing the negative under the square root, \(-1\), can be expressed as \(i^2 = -1\). By rewriting, \(\sqrt{-39} = \sqrt{-1 \times 39} = i \sqrt{39}\). Incorporating this into the original expression, we have \(-\sqrt{-39} = -i\sqrt{39}\). This simplification process not only makes the expression easier to work with but also ensures that it is in the simplest form that adheres to the rules of complex numbers.