Problem 54 - Algebra for College Students

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. For example, $$ \begin{aligned} \sqrt{-3} \sqrt{-8} &=(i \sqrt{3})(i \sqrt{8}) \\ &=i^{2} \sqrt{24} \\ &=(-1) \sqrt{4} \sqrt{6} \\ &=-2 \sqrt{6} \end{aligned} $$ $$\sqrt{-75} \sqrt{3}$$

Step-by-Step Solution

Answer

\( 15i \)

Step 1: Express Each Radicand with 'i'

Start by recognizing that the expression \( \sqrt{-75} \sqrt{3} \) involves a negative radicand in \( \sqrt{-75} \). Rewrite it using \( i \) as follows: \( \sqrt{-75} = i \sqrt{75} \). This results in the expression: \( \sqrt{-75} \sqrt{3} = (i \sqrt{75}) \sqrt{3} \).

Step 2: Multiply the Radicals

Multiply the radicals under a single square root: \( (i \sqrt{75}) \sqrt{3} = i \sqrt{75 \times 3} \). This gives: \( i \sqrt{225} \).

Step 3: Simplify the Radical Expression

Simplify \( \sqrt{225} \): \( i \sqrt{225} = i \times 15 = 15i \).

Step 4: Final Simplified Form

Conclude by stating the simplified form of the original expression is \( 15i \).

Key Concepts

Imaginary Unit (i)

The imaginary unit, commonly denoted by the letter \(i\), is a fundamental concept in dealing with complex numbers. It represents the square root of \(-1\), an idea that might initially seem strange since no real number squared will give a negative result. Here's what you need to understand about \(i\):

• Definition: \(i = \sqrt{-1}\).
• Property: Since \(i = \sqrt{-1}\), it logically follows that \(i^2 = -1\).

The imaginary unit allows us to handle square roots of negative numbers by providing a solution in terms of complex numbers. In our exercise, when we deal with \(\sqrt{-75}\), we replace it with \(i\sqrt{75}\) to bypass the issue of taking the square root of a negative radicand.

Negative Radicand

A radicand is the number or expression inside a radical symbol, and when it's negative, it presents a conceptual challenge. Let's break down what to do when you encounter a negative radicand:

• Identifying Negative Radicands: In expressions like \(\sqrt{-75}\), the negative sign inside the square root indicates a negative radicand.
• Utilizing Imaginary Units: Convert this problematic expression into a form that includes \(i\) by rewriting \(\sqrt{-75}\) as \(i\sqrt{75}\). This transformation makes calculations manageable.

By applying imaginary units, we easily extend our arithmetic operations to include values which would otherwise be undefined in the real number system.

Simplifying Radicals

Simplifying radicals is a valuable skill in mathematics, especially when working with complex numbers and expressions that involve radical combinations. Let's go through how to simplify radicals, focusing on our current problem:

• Combining Radicals: We first transform \(\sqrt{-75} \times \sqrt{3}\) into \(i \sqrt{75 \times 3}\), which becomes \(i \sqrt{225}\).
• Simplifying the Expression: After computing \(\sqrt{225}\), which equals \(15\), we multiply it by \(i\) to finalize the expression. Thus, \(i \times 15 = 15i\).

These steps demonstrate the ease of simplifying radicals by combining them under a single radical when possible, making calculations straightforward and tidy.