Problem 71 - Intermediate Algebra

Question

Find each of the products and express the answers in the standard form of a complex number. $$ (6-2 i)(7-i) $$

Step-by-Step Solution

Answer

The product is \(40 - 20i\).

Step 1: Expand the Product

To find the product \((6-2i)(7-i)\), we start by using the distributive property (FOIL method):- First: Multiply 6 by 7 to get 42.- Outer: Multiply 6 by \(-i\) to get \(-6i\).- Inner: Multiply \(-2i\) by 7 to get \(-14i\).- Last: Multiply \(-2i\) by \(-i\) to get \(2i^2\).

Step 2: Simplify and Substitute

Calculate the resultant terms using each multiplication:- The result of multiplying \(i^2\) is \(-1\), so \(2i^2 = 2(-1) = -2\).- Combine all the terms: \(42 - 6i - 14i - 2\).

Step 3: Combine Like Terms

- Simplify the expression further by combining the real and imaginary parts:- Real parts: \(42 - 2 = 40\).- Imaginary parts: \(-6i - 14i = -20i\).

Step 4: Present in Standard Form

The final complex number in standard form is the combination of the simplified real and imaginary terms:- The result is \(40 - 20i\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept used to simplify expressions involving multiplication over addition or subtraction. When dealing with complex numbers, this property helps in breaking down products into manageable parts.

Consider multiplying two binomials, such as \(6 - 2i\) and \(7 - i\). You distribute each term in the first binomial across each term in the second. This method allows for more straightforward multiplication of complex numbers.

• Multiply the first terms in each binomial together.
• Continue by multiplying the outer and inner terms.
• Finish by multiplying the last terms.

By breaking it down this way, you can easily combine like terms after calculating each simple multiplication. This property is crucial when manipulating expressions containing complex numbers, as seen in the exercise.

FOIL Method

The FOIL method is an acronym that guides you through the process of multiplication for two binomials: First, Outer, Inner, Last. This method ensures you do not leave out any terms when using the distributive property.

For example, in the multiplication of \(6 - 2i\) and \(7 - i\):

• **First**: Multiply the first terms of each binomial, such as \(6\) and \(7\).
• **Outer**: Multiply the outer terms, \(6\) and \(-i\).
• **Inner**: Multiply the inner terms, \(-2i\) and \(7\).
• **Last**: Finally, multiply the last terms, \(-2i\) and \(-i\).

After applying these steps, simplify by combining like terms. This method helps to maintain accuracy and ensures that each part of the distribution is covered, facilitating easier manipulation of the expressions involved.

Imaginary Numbers

Imaginary numbers are a class of numbers that include the imaginary unit 'i', which is defined as the square root of \(-1\). Imaginary numbers expand real numbers to form complex numbers consisting of a real and an imaginary part.

When handling operations involving imaginary numbers, it's crucial to remember basic rules:

• \(i^2 = -1\) is the fundamental identity of imaginary numbers.
• Imaginary numbers are often seen in expressions like \(-2i\), where 'i' represents the imaginary part.

In our example, during multiplication, you'll encounter \(i^2\) which converts to \(-1\). Such transformations are key in simplifying complex number expressions and are commonplace in operations involving imaginary numbers.

Standard Form of a Complex Number

Every complex number can be expressed in a standard form of \(a + bi\), where 'a' is the real part, and 'b' is the coefficient of the imaginary part. This form gives a clear separation between real and imaginary components.

For the multiplication example \(6 - 2i)(7 - i)\), once all computations and combinations of real and imaginary terms are performed, the result should be expressed in standard form.

• After simplification, the real part is \(40\).
• The imaginary part becomes \(-20i\).

Thus, the final answer is \(40 - 20i\), a clear representation using the standard form. This uniform format helps in further computations and makes interpreting the results straightforward.