Problem 46 - Intermediate Algebra

Question

Multiply. $$ \sqrt{-9} \cdot \sqrt{-100} $$

Step-by-Step Solution

Answer

The product is -30.

Step 1: Identify the Numbers

Note that each square root is of a negative number: \( \sqrt{-9} \) and \( \sqrt{-100} \). These involve imaginary numbers, as square roots of negative numbers are not defined in the real number system.

Step 2: Express Each as an Imaginary Number

Recall that the square root of a negative number can be expressed using \( i \), where \( i = \sqrt{-1} \). So, \( \sqrt{-9} = \sqrt{9} \cdot \sqrt{-1} = 3i \) and \( \sqrt{-100} = \sqrt{100} \cdot \sqrt{-1} = 10i \).

Step 3: Multiply the Imaginary Numbers

Multiply the two expressions together: \( (3i) \cdot (10i) = 30i^2 \).

Step 4: Simplify Using the Property of Imaginary Units

Since \( i^2 = -1 \), substitute \( -1 \) for \( i^2 \) in the expression: \( 30i^2 = 30(-1) = -30 \).

Key Concepts

Complex Numbers

Complex numbers are an extension of the regular numbers we use every day, called real numbers. They include both a real part and an imaginary part. The general form of a complex number is written as

• \( a + bi \)
• where \( a \) is the real part and \( bi \) is the imaginary part
• and \( i \) represents the imaginary unit, defined as \( \sqrt{-1} \)

The imaginary unit \( i \) is what allows us to work with the square roots of negative numbers, a concept not possible within the realm of real numbers. By convention, when you are dealing with complex numbers, expressions like \( \sqrt{-9} \), which involve negative values under the square root, are considered undefined unless expressed in terms of imaginary numbers. This leads us naturally into understanding how square roots behave in this new set of numbers.

Square Roots

Square roots are functions that help determine what number, when multiplied by itself, will equal the original number. You may already be comfortable taking square roots of positive numbers. For example

• the square root of 9 is 3 because \( 3 \times 3 = 9 \)
• For negative numbers, however, square roots initially seem perplexing because there is no real number that can multiply by itself to yield a negative number.

This brings in the role of imaginary numbers. For instance

• vigorous understanding of these concepts allows expressions like \( \sqrt{-9} \)
• to be transformed using the identity \( i = \sqrt{-1} \)
• to become \( 3i \)

This step is crucial in problems like those involving multiplying complex numbers, as it sets the stage for further operations.

Multiplying Complex Numbers

When you multiply complex numbers, you will treat the imaginary unit \( i \) like a variable. Remember when multiplying like terms, similar to variables in algebra, the symbols will follow their respective rules. Consider the exercise of multiplying the results of two square roots

• \( (3i) \times (10i) \)
• Here you multiply each part: the real number parts (3 and 10) together, and treat \( i \times i \) as \( i^2 \)
• This gives \( 30i^2 \)

Since \( i^2 = -1 \), substitute it into the product to reach the simplified result. This means \( 30i^2 \) converts to \(-30\). Remember that the operation with the imaginary parts transforms to a real number during this multiplication process. This is a key feature of working with complex numbers, as it allows these once mysterious portions to interact sensibly with the arithmetic properties you're familiar with.