Problem 49 - Intermediate Algebra

Question

$$ (7 i)(-9 i) $$

Step-by-Step Solution

Answer

63

Step 1: Understand the Variables

In this expression, we have two imaginary numbers multiplied together: \(7i\) and \(-9i\). Where \(i\) is the imaginary unit, defined as \(i=\sqrt{-1}\).

Step 2: Apply the Multiplication Rule

Multiply the coefficients of the imaginary parts first: \(7\times (-9) = -63\).

Step 3: Multiply the Imaginary Units

The multiplication of the imaginary units is given by: \(i\times i = i^2\).

Step 4: Use the Definition of \(i^2\)

Recall that \(i^2 = -1\). Substitute \(i^2\) with \(-1\) in the expression: \(-63\times i^2 = -63\times (-1)\).

Step 5: Perform the Final Calculation

Simplify the expression by multiplying the numbers: \(-63\times (-1) = 63\).

Step 6: Arrive at the Final Answer

The product of \((7i)\times (-9i)\) is calculated to be \(63\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental building block in complex numbers. It is defined by the property \(i = \sqrt{-1}\). This definition helps to simplify calculations involving square roots of negative numbers, which are not possible within the set of real numbers. The introduction of \(i\) expands the real number system to include complex numbers, thereby broadening the field of numbers to handle mathematical and engineering problems that involve such roots.

• Imaginary numbers are numbers that can be written in the form of \(bi\), where \(b\) is a real number and \(i\) is the imaginary unit.
• Complex numbers are expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers.

Understanding \(i\)'s definition is crucial for working with complex numbers, as it forms the basis for all operations involving imaginary components.

Multiplication of Complex Numbers

Multiplying complex numbers involves using the distributive property just as you would with binomials. Let's look at the example of multiplying \((7i)\) and \((-9i)\).

• First, multiply their coefficients: \(7\times (-9) = -63\).
• Next, multiply the imaginary units: \(i\times i = i^2\).

This step shows how operations with complex numbers are an extension of algebraic manipulation involving real numbers.
Now you incorporate the property of \(i^2\) and reform the expression:

• The expression simplifies to \(-63\times i^2\).
• Since \(i^2 = -1\), replace \(i^2\) by \(-1\): \(-63\times (-1)\).
• Finally, calculate the result: \(-63 imes (-1) = 63\).

By breaking down the multiplication into manageable parts, it becomes easier to handle these seemingly complex calculations.

Properties of i

The properties of \(i\) revolve around its unique standing as the imaginary unit. These properties are essential to performing operations within the set of complex numbers. Let's explore them in detail:

• By definition, \(i = \sqrt{-1}\).
• The square of the imaginary unit, \(i^2 = -1\), reverses the sign when multiplied into any expression.
• Continuing to multiply \(i\) by itself leads to a repeating cycle: \(i^3 = i \times i^2 = i \times (-1) = -i\) and by \(i^4 = i^3 \times i = -i \times i = 1\).
• This cycle repeats, which means that for any integer power \(n\), you can simplify \(i^n\) based on the remainder when \(n\) is divided by 4.

These properties help in simplifying and evaluating powers of \(i\) within expressions, making calculations more efficient when dealing with complex numbers.