Problem 67 - Intermediate Algebra

Question

Perform each indicated operation. Write the result in the form \(a+b i\). $$ (9+8 i)^{2} $$

Step-by-Step Solution

Answer

The result is \(17 + 144i\).

Step 1: Expand the Square

To find \((9+8i)^2\), we need to square the binomial. Use the formula for squaring a binomial, \((a+b)^2 = a^2 + 2ab + b^2\):\[(9 + 8i)^2 = 9^2 + 2(9)(8i) + (8i)^2\]

Step 2: Calculate Each Term Separately

Calculate the square of each term and the product:- \(9^2 = 81\)- The middle term: \(2(9)(8i) = 144i\)- \((8i)^2 = 64i^2\)Recall that \(i^2 = -1\), so \(64i^2 = 64(-1) = -64\).

Step 3: Combine the Real and Imaginary Parts

Combine all the calculated terms together:- Real parts: \(81 - 64\)- Imaginary part: \(144i\)Hence, the combined expression is:\[(81 - 64) + 144i = 17 + 144i\]

Step 4: Write the Final Result

The expression after performing all operations becomes \(17 + 144i\). This is in the form of \(a + bi\).

Key Concepts

Binomial Expansion

The binomial expansion is a method used to expand expressions that are raised to a power. For example, if you have an expression like \((9 + 8i)^2\), the binomial expansion helps to simplify and solve it. It's important to understand that binomials consist of two terms, thus why the name "bi-" meaning two.

When you expand a binomial square \((a + b)^2\), you apply the formula: \(a^2 + 2ab + b^2\). This formula allows you to simplify the expression by calculating individual terms separately:

• \(a^2\): Square the first term.
• \(2ab\): Multiply the first term, the second term, then multiply by 2.
• \(b^2\): Square the second term.

This is the same process used in our exercise: \((9 + 8i)^2\) becomes \(9^2 + 2(9)(8i) + (8i)^2\). Each component is calculated separately to provide a clean solution.

Imaginary Unit

The imaginary unit, denoted by \(i\), is a crucial element in complex numbers. It is defined by its property that \(i^2 = -1\). This concept is essential when dealing with imaginary numbers, which are numbers that can be written as a real number multiplied by \(i\).

In complex number operations, the imaginary unit allows us to handle expressions that involve the square root of negative numbers. For example, multiplying \(i\) by itself results in \(i^2 = -1\), as seen in the term \((8i)^2 = 64i^2 = 64(-1) = -64\). This transformation is key when simplifying expressions with imaginary parts.

Real and Imaginary Parts

In the final expression of a complex number, such as \(17 + 144i\), it is essential to distinguish between the real and imaginary parts. A complex number is generally expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) represents the coefficient of the imaginary part.

In our example, the real part is 17, which comes from summing the calculated real components (\(81\) and \(-64\)). The imaginary part, 144i, originates from the calculated imaginary component. Being able to separate these parts is vital in order to understand and simplify complex expressions, and perform algebraic operations correctly.

Algebraic Operations

Algebraic operations with complex numbers involve addition, subtraction, multiplication, and division, treating imaginary numbers with special considerations. These operations can often be broken down by handling the real and imaginary parts separately at each step.

For instance, when multiplying binomials such as in \((9+8i)^2\), you follow algebraic principles to expand and simplify, while respecting properties like \(i^2 = -1\). Addition and subtraction within complex numbers involve combining the like parts—real with real, imaginary with imaginary.

• Add real numbers separately from the imaginary numbers.
• Multiply real parts and imaginary parts according to algebraic rules.
• Simplify using the property of the imaginary unit—\(i^2 = -1\)—where applicable.

This ensures that complex expressions are simplified accurately, as seen when \(81 - 64\) yields the real part and \(144i\) remains as the imaginary part.