Problem 49 - Algebra for College Students

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\sqrt{-15} \sqrt{-5}$$

Step-by-Step Solution

Answer

\(-5\sqrt{3}\)

Step 1: Express in Terms of i

Start by expressing each square root in terms of the imaginary unit, \(i\). The imaginary unit is defined as \(i = \sqrt{-1}\). Therefore, \(\sqrt{-15} = i \sqrt{15}\) and \(\sqrt{-5} = i \sqrt{5}\).

Step 2: Multiply the Expressions

Perform the multiplication of the two expressions we found in Step 1. Multiply \( (i \sqrt{15})(i \sqrt{5}) \).

Step 3: Simplify Using i^2

Recognize that \( i^2 = -1 \). So, \( (i \sqrt{15})(i \sqrt{5}) = i^2 \sqrt{75} = -\sqrt{75} \).

Step 4: Simplify the Square Root

Simplify \( \sqrt{75} \). Since \( 75 = 25 \times 3 \), we have \( \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \sqrt{3} = 5 \sqrt{3} \).

Step 5: Final Result

Combine the result from Steps 3 and 4 to express the final answer: \( -\sqrt{75} = -5\sqrt{3} \).

Key Concepts

Imaginary Unit

When dealing with complex numbers, one of the most important concepts is the imaginary unit, denoted as \( i \). At its core, \( i \) is defined as \( \sqrt{-1} \).

This means that whenever you encounter a square root of a negative number, you can express it using \( i \).

• For example, \( \sqrt{-4} \) is expressed as \( 2i \) because it equals \( \sqrt{4} \cdot \sqrt{-1} = 2 \cdot i \).
• Similarly, \( \sqrt{-9} = 3i \).

By using \( i \), we can transform negative square roots into purely imaginary parts. Understanding this will enable you to handle and simplify even more complex expressions involving negative roots.

Simplifying Square Roots

Simplifying square roots is crucial when working with any type of number, including complex numbers. The process involves breaking down a square root into its simplest form.

For a positive square root, like \( \sqrt{75} \), we find factors that are perfect squares.

• Identify that 75 can be factored into 25 and 3, with 25 being a perfect square.
• This means \( \sqrt{75} = \sqrt{25 \times 3} \), which simplifies to \( 5\sqrt{3} \).

The same principle applies to more complex expressions, allowing us to express them in a much simpler and more usable form. This step is essential before any multiplication or addition involving square roots, as it simplifies the calculations significantly.

Multiplying Complex Numbers

Multiplying complex numbers involves not only multiplying their real and imaginary parts but also applying the fundamental identity \( i^2 = -1 \).

For instance, when multiplying two complex numbers (e.g., \((i \sqrt{15})(i \sqrt{5}) \)), follow these steps:

• Multiply the coefficients and the transitive \( i \)'s: \( i \cdot i = i^2 \), which equals \(-1\).
• This results in \( i^2 \sqrt{75} = -\sqrt{75} \).

Using the identity \( i^2 = -1 \) not only simplifies expressions but also switches the sign from positive to negative, changing the nature of the number. This conversion leads to a real number component which might be negative, depending on the initial complex expressions. This step ensures the correct simplification of the outcome after multiplication.