Problem 66 - A Graphical Approach to College Algebra

Question

Multiply as indicated. Write each product in standard form. $$(\sqrt{2}-4 i)(\sqrt{2}+4 i)$$

Step-by-Step Solution

Answer

The product is 18 in standard form.

Step 1 - Identify the Conjugates

Notice that the expression consists of two binomials, \( (\sqrt{2} - 4i) \) and \( (\sqrt{2} + 4i) \), that are conjugates of each other. Binomial conjugates take the form \((a + bi)(a - bi)\).

Step 2 - Apply the Difference of Squares Formula

When multiplying conjugates \((a + bi)(a - bi)\), the result is \( a^2 + b^2 \), since the middle terms cancel each other out. Here, \(a = \sqrt{2}\) and \(b = 4\).

Step 3 - Calculate \(a^2\)

Square the real part \(a = \sqrt{2}\): \[ (\sqrt{2})^2 = 2 \]

Step 4 - Calculate \(b^2\)

Square the imaginary coefficient \(b = 4\): \[ 4^2 = 16 \]

Step 5 - Add \(a^2\) and \(b^2\)

Add the results from Step 3 and Step 4 to find the final result:\[ 2 + 16 = 18 \]

Step 6 - Write the Product in Standard Form

The product of the conjugates \((\sqrt{2} - 4i)(\sqrt{2} + 4i)\) is \(18\), which is a real number, thus already in standard form.

Key Concepts

Binomial Conjugates

Complex numbers are numbers that consist of both a real part and an imaginary part and can be conveniently represented as binomials. In the exercise, we encounter binomials such as \(\sqrt{2} - 4i\) and \(\sqrt{2} + 4i\). These two binomials are known as conjugates. Conjugates have a special form: when you multiply them, the imaginary components cancel each other out.

• A conjugate is simply formed by changing the sign between the real part and the imaginary part. For example, if one binomial is \(a + bi\), its conjugate would be \(a - bi\).
• In our case, \(a = \sqrt{2}\) and \(b = 4\), so \((\sqrt{2} - 4i)\) and \((\sqrt{2} + 4i)\) are conjugates of each other.

This cancellation of imaginary parts makes conjugates particularly useful, as the resulting product is a real number, simplifying calculations involving complex numbers.

Difference of Squares

When dealing with conjugates, we utilize the difference of squares formula. The difference of squares is a pattern that occurs when multiplying two conjugates together.

• The formula states that for any two numbers \(a\) and \(b\), the product \((a + b)(a - b)\) is \(a^2 - b^2\).
• But in the case of complex conjugates like \((a + bi)(a - bi)\), the formula modifies slightly to become \(a^2 + b^2\).

In the exercise, here’s how these squares work:

• Square the real number: \((\sqrt{2})^2 = 2\).
• Square the imaginary part’s coefficient: \(4^2 = 16\).

Then, simply sum these squares: \(2 + 16 = 18\), which is also the final product, already simplified to a real number.

Standard Form in Algebra

In algebra, it's common to express complex numbers in a standard form, which is a way of writing them clearly. The standard form for a complex number is \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.Let's breakdown what this means:

• If a complex number is already simplified to a real number, like \(18\) in our exercise, that's still considered its standard form but as a purely real number. This is because the imaginary part \(bi\) is equal to zero.
• It is crucial to recognize what's considered standard, especially when multiplying complex numbers, so that the answer is easily understandable.

In this problem, after utilizing the properties of conjugates and the difference of squares, we reach a solution that is a real number, \(18\). Although there is no imaginary part, it is perfectly acceptable and expected for it to be in this format.