Problem 10 - Precalculus: Functions and Graphs

Question

Write the expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$(6+7 i)^{2}$$

Step-by-Step Solution

Answer

The expression is \(-13 + 84i\).

Step 1: Identify Format for Squaring

We need to write \((6 + 7i)^2\) in the form \(a + bi\). This requires expanding the expression using the formula \((x+y)^2 = x^2 + 2xy + y^2\).

Step 2: Expand the Expression

Let's expand \((6 + 7i)^2\) using the formula. Here, \(x = 6\) and \(y = 7i\). So, \((6 + 7i)^2 = 6^2 + 2 \times 6 \times 7i + (7i)^2\).

Step 3: Calculate Each Component

First, calculate \(6^2 = 36\). Next, calculate \(2 \times 6 \times 7i = 84i\). Finally, calculate \((7i)^2 = 49i^2 = 49(-1) = -49\) (because \(i^2 = -1\)).

Step 4: Combine Real and Imaginary Parts

Now combine the calculated parts from Step 3:\(36 + 84i - 49\). Combine the real terms: \(36 - 49 = -13\).Thus, the expression becomes \(-13 + 84i\).

Step 5: Write in the Form \(a + bi\)

The expression is now in the required form: \(a = -13\), \(b = 84\), thus \(-13 + 84i\).

Key Concepts

Imaginary Unit

In mathematics, particularly in complex number theory, the imaginary unit is denoted by the letter 'i'. It holds a unique property as it is defined to be the square root of -1. This allows for the extension of real numbers into the complex number system, enabling operations that would otherwise yield no result within the real numbers. For example, if we're faced with finding the square root of a negative number, we can utilize the imaginary unit:

• The square of the imaginary unit is \[ i^2 = -1 \].
• This means \[ i = \sqrt{-1} \].

Recognizing this is crucial in operations involving complex numbers, as evidenced when squaring complex numbers like \( (6 + 7i)^2 \). Here, understanding \( i^2 \) is necessary to simplify and combine like terms correctly after expansion.

Expansion of Binomials

The process of expanding binomials, like \( (x + y)^2 \), involves applying the binomial theorem or a specialized formula for squaring. In our particular problem, we expanded \( (6 + 7i)^2 \) by using the formula:

• \[ (x+y)^2 = x^2 + 2xy + y^2 \]

This formula is a handy tool because it allows for a systematic method to handle the cross-product terms that appear in the square of a binomial. By substituting \( x = 6 \) and \( y = 7i \) into the formula, we can find each term that contributes to the expanded expression:

• First, compute \( x^2 = 6^2 = 36 \).
• Next, the cross-product term \( 2xy = 2 \times 6 \times 7i = 84i \).
• Finally, \( y^2 = (7i)^2 = 49i^2 \), which simplifies to \(-49 \).

Each step follows logically from applying the expansion formula, helping to transform a complex expression elegantly into a more usable form, \( a + bi \).

Squaring Complex Numbers

When dealing with complex numbers, such as squaring \((6 + 7i)^2\), it's crucial to handle both the real and imaginary parts correctly. Each component—whether part of the real number or the imaginary 'i' term—must be carefully calculated and combined:

• Start by squaring each part separately, as shown in the expansion: \(6^2 = 36\) and \((7i)^2 = 49i^2\).
• Remember the property \(i^2 = -1\), which flips the sign of the result when squaring an imaginary number. Thus, \(49i^2 = 49 \times (-1) = -49 \).
• Don’t forget the middle term from the expansion (the cross-product), which is \(84i\), contributing entirely as an imaginary component.
• Lastly, combine the real components \(36\) and \(-49\) to find the final result: \(-13\).

This combination of real and imaginary is how we achieve the final form \(-13 + 84i\). Squaring complex numbers thereby requires careful attention to maintain both the structural and numeric integrity of the resulting expression.