Problem 24 - A Graphical Approach to College Algebra

Question

Write each number in simplest form, without a negative radicand. $$-\sqrt{-225}$$

Step-by-Step Solution

Answer

The simplest form is \(-15i\).

Step 1: Rationalize the Negative Radicand

The expression given is \(-\sqrt{-225}\). First, notice that there is a negative sign inside the square root, indicating an imaginary number. This can be written as \(-\sqrt{225} \times \sqrt{-1}\) which is \(-\sqrt{225} \cdot i\), where \(i\) is the imaginary unit.

Step 2: Simplify the Square Root

Next, simplify \(\sqrt{225}\). Recognize that \(225\) is a perfect square. In fact, \(\sqrt{225} = 15\), so substitute this value to get \(-15 \cdot i\).

Step 3: Express in Simplest Form

Finally, simplify the expression. The simplest form of the expression with imaginary numbers is \(-15i\). This combines the previously found square root and imaginary unit into one term.

Key Concepts

Complex Numbers

Complex numbers are fascinating because they extend our number system beyond the real numbers you often work with.
In this system, a complex number has a real and an imaginary part, expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The imaginary unit \(i\) is defined as the square root of \(-1\), meaning that \(i^2 = -1\). This lets us easily handle square roots of negative numbers.
In the given exercise, \(-15i\) is a complex number because it includes \(i\), indicating the imaginary part. Although the real part \(a\) is zero in this case, complex numbers often blend real and imaginary parts.
Understanding complex numbers is crucial as they frequently appear in advanced mathematics, physics, and engineering for solving equations that have no real solutions.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots.
Working with radicals, in particular, when they include negative numbers involves recognizing the role of imaginary numbers.
For a square root of a negative number, as seen in the exercise, you transform the expression to include the imaginary unit \(i\).

• For instance, the expression \(-\sqrt{-225}\) is broken down to \(-\sqrt{225} \cdot \sqrt{-1}\).
• This translates to \(-\sqrt{225} \cdot i\), simplifying to \(-15i\).

Through this process, radicals that feature negative radicands are simplified, making them more manageable in complex number problems. The introduction of \(i\) allows these expressions to be evaluated systematically.

Simplifying Radicals

Simplifying radicals is an important skill in math, helping you express numbers in their most basic form.
When simplifying a radical, your goal is to find an equivalent expression where the radicand (the number under the root) is as small as possible.
In the exercise where you have \(-\sqrt{-225}\), the process involves a few steps:

• First, identify the perfect square within the radicand, such as \(225\), simplifying it to \(15\).
• Then acknowledge the negative by involving the imaginary unit \(i\), as the square root of \(-1\).
• This reduces the given expression to the simplest form of \(-15i\).

By simplifying radicals effectively, you achieve a cleaner, more understandable form,
which is especially useful when you encounter complex equations or expressions in higher-level math.