Problem 108 - Intermediate Algebra

Question

Write in the form \(a+b i\). $$ 5-\sqrt{-16} $$

Step-by-Step Solution

Answer

The expression is \(5 - 4i\).

Step 1: Identify the imaginary component

The term \(-\sqrt{-16}\) needs to be expressed in terms of the imaginary unit \(i\). Recall that the square root of \(-1\) is \(i\). Thus, \(-\sqrt{-16}\) can be rewritten as \(-\sqrt{16} \cdot \sqrt{-1}\).

Step 2: Simplify the square root

Calculate \(\sqrt{16}\), which is \(4\). Therefore, \(-\sqrt{-16}\) becomes \(-4i\) because the imaginary unit \(\sqrt{-1}\) is represented as \(i\).

Step 3: Combine with the real term

Now, combine the real number \(5\) with the simplified imaginary term \(-4i\). This gives you the expression \(5 - 4i\).

Key Concepts

Understanding the Imaginary Unit

When dealing with complex numbers, the imaginary unit is a fundamental part. The imaginary unit is denoted as \(i\), and it is defined as the square root of \(-1\). This might sound confusing initially because normally, the square root of a negative number isn't a real number. However, in the world of complex numbers, \(i\) is the tool that allows us to handle these situations.

• \(i^2 = -1\): This helps when performing operations in complex arithmetic.
• Imaginary numbers are expressed as multiples of \(i\).
• In expressions, \(i\) typically follows the numeric coefficient.

Recognizing \(i\) as the basis for imaginary numbers helps simplify complex expressions by identifying parts that incorporate \(\sqrt{-1}\). This understanding is crucial for operations involving complex numbers.

Working with Square Roots of Negative Numbers

Square roots are numbers that, when multiplied by themselves, yield the original number. In the case of positive numbers, this is straightforward. For example, the square root of 16 is 4 because \(4 \times 4 = 16\).
But, what happens when you have to calculate the square root of a negative number, like \(-16\)? This is where complex numbers—and the imaginary unit \(i\)—come into play.

• The square root of a negative number can be expressed as a product of the square root of its positive counterpart and the square root of \(-1\): \(\sqrt{-16} = \sqrt{16} \times \sqrt{-1}\).
• This simplifies to \(4 \cdot i = 4i\).

The original negative sign still affects the overall product. Thus, if you have \(-\sqrt{-16}\), it translates to \(-4i\), combining the negative effect with the imaginary component.

Simplifying Expressions with Real and Imaginary Terms

Simplifying expressions involving complex numbers usually involves combining real and imaginary parts. In the exercise given, you start with a real number, 5, and subtract an imaginary component, \(-4i\). To simplify, you perform the subtraction as follows:

• The real number remains the same because there's no real counterpart to subtract from it.
• The imaginary term is simply \(-4i\) because it contains the imaginary unit \(i\).
• Therefore, combining these gives the simplified expression \(5 - 4i\).

When simplifying, it helps to see complex numbers as a composite of real and imaginary parts: \(a + bi\). This distinction makes calculations clear and ensures that you properly handle each part of the expression. The result, \(5 - 4i\), illustrates how both components integrate into a single complex number form.