Problem 44 - Intermediate Algebra

Question

Write each of the following in terms of \(i\), perform the indicated operations, and simplify. $$ \sqrt{-81} \sqrt{-25} $$

Step-by-Step Solution

Answer

The simplified expression is -45.

Step 1: Understand the Problem

We need to simplify the expression \( \sqrt{-81} \sqrt{-25} \), which involves imaginary numbers. We will express each square root in terms of \( i \), where \( i = \sqrt{-1} \).

Step 2: Express Each Term in Terms of i

First, express \( \sqrt{-81} \) and \( \sqrt{-25} \) with \( i \):\[ \sqrt{-81} = \sqrt{81} \cdot \sqrt{-1} = 9i \]\[ \sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i \]

Step 3: Multiply the Terms

Multiply the expressions \( 9i \) and \( 5i \):\[ 9i \times 5i = 45i^2 \]

Step 4: Simplify Using the Property of i²

Since \( i^2 = -1 \), substitute \( i^2 \) with \(-1\) in the expression:\[ 45i^2 = 45(-1) = -45 \]

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real part and an imaginary part. The imaginary unit is denoted as \( i \), where \( i = \sqrt{-1} \). This concept becomes essential when dealing with the square roots of negative numbers. Complex numbers can take the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. In the context of the original problem, \( 9i \) and \( 5i \) are purely imaginary numbers, having a real part of zero.

Understanding complex numbers helps in simplifying expressions that involve square roots of negative numbers. When you multiply two imaginary terms like \( 9i \) and \( 5i \), you must remember the property of \( i^2 \). This property, \( i^2 = -1 \), is pivotal in further simplifying expressions into real numbers.

Square Roots

Square roots often appear simple, but they become a bit more complex when involving negative numbers. In standard mathematical operations, you can't directly take the square root of a negative number using real numbers. This is where the imaginary unit \( i \) is beneficial. In the given expression \( \sqrt{-81} \sqrt{-25} \), by understanding the rule \( \sqrt{-n} = \sqrt{n} \cdot i \), you convert negative square roots into forms you can manage.

For instance:

• \( \sqrt{-81} \) becomes \( 9i \) because \( \sqrt{81} = 9 \), and we multiply it with \( i \).
• \( \sqrt{-25} \) similarly becomes \( 5i \), using the same logic \( \sqrt{25} = 5 \).

By rewriting the square roots of negative numbers in terms of \( i \), you transform them into manageable components for further operations.

Simplifying Expressions

Simplifying expressions, especially in algebra, means reducing them to their simplest form. When you multiply imaginary numbers like \( 9i \times 5i \), it translates into \( 45i^2 \) which initially may look complex. But using the property \( i^2 = -1 \), you can further simplify this expression.

Let's simplify step by step:

• First, calculate the multiplication \( 9i \times 5i = 45i^2 \).
• Next, substitute \( i^2 \) with \(-1\), giving you \( 45i^2 = 45 \times -1 \).
• Finally, simplify to obtain \( -45 \).

Through these simplification steps, you convert what initially seemed a complicated expression back into a simple real number. This showcases how foundational concepts like the imaginary unit \( i \) and the operation properties of imaginary numbers are utilized to achieve simplification.