Problem 20 - Precalculus Student Solutions Manual 5th

Question

Write each number as the product of a real number and i. $$\sqrt{-15}$$

Step-by-Step Solution

Answer

\(\sqrt{15} i\)

Step 1 - Understand the imaginary unit

The imaginary unit is denoted as \(i\) and is defined as \(i = \sqrt{-1}\).

Step 2 - Break down the square root

Given \(\root{\sqrt{-15}}\), we can rewrite this as \(\sqrt{15} \cdot \sqrt{-1}\).

Step 3 - Use the imaginary unit

Since \(\sqrt{-1} = i\), we substitute the value of \(\sqrt{-1}\) with \(i\). Thus, \(\sqrt{15} \cdot i\) becomes the product of the real number \(\sqrt{15}\) and \(i\).

Key Concepts

imaginary unit

The imaginary unit is a fundamental concept in mathematics, especially in complex numbers. Denoted by the symbol \(i\), it is defined as \(i = \sqrt{-1}\). This definition essentially allows us to handle the square root of negative numbers, which are not possible within the realm of real numbers. Imaginary numbers, therefore, extend our number system.
For example, if we take the square root of negative one: \(i \cdot i = (\sqrt{-1}) \cdot (\sqrt{-1}) = -1\). Thus, \(i^2 = -1\) always holds true.
Imaginary numbers are crucial in complex number theory and find applications in various fields such as engineering, physics, and even signal processing.

square root of negative numbers

When we encounter the square root of a negative number, the imaginary unit \(i\) comes into play. Negative numbers do not have real square roots because no real number squared gives a negative result.
For example, consider \(\sqrt{-15}\). To handle this, we split the process into two parts:

• First, identify the positive part of the number, \(\sqrt{15}\).
• Second, use the imaginary unit \(\sqrt{-1} = i\).

Combining these, we have \(\sqrt{-15} = \sqrt{15} \cdot i\). This method transforms the problem into a product involving a real number and the imaginary unit.

product involving i

Multiplying with the imaginary unit \(i\) is straightforward once you understand the fundamental properties of \(i\).
For instance, if we need to write \(\sqrt{-15}\) as a product involving \(i\), we perform the following steps:

• First, recognize that \(\sqrt{-15}\) can be broken down into \(\sqrt{-1}\) and \(\sqrt{15}\).
• Second, substitute \(\sqrt{-1}\) with \(i\).
• This gives us \(\sqrt{15} \cdot i\).

Therefore, \(\sqrt{-15}\) rewritten as a product of a real number and \(i\ is \sqrt{15} \cdot i\). This format allows us to handle complex numbers efficiently and express solutions clearly.