Problem 46 - College Algebra

Question

Perform the indicated operation(s) and write the result in standard form. $$ (8+9 i)(2-i)-(1-i)(1+i) $$

Step-by-Step Solution

Answer

The result of the operation is: \(7 + 10i\).

Step 1: Multiply the first two complex numbers

Multiply \(8+9i\) and \(2-i\) by applying the distributive law. You obtain: \[ (8*2) + (8*-i) + (9i*2) + (9i*-i) = 16 - 8i + 18i - 9 \]

Step 2: Combine like terms in the first multiplication

Combine the real and imaginary parts from the result of the first multiplication: \[(16 - 9) + (-8i + 18i) = 7 + 10i \]

Step 3: Multiply the last two complex numbers

Multiply \(1-i\) and \(1+i\) by applying the distributive law. Here, you obtain: \[(1*1) + (1*i) + (-i*1) + (-i*i) = 1 + i - i - 1 = 0\]

Step 4: Subtract the result of Step 3 from the result of Step 2

Subtract the result obtained in Step 3 from the result obtained in step 2. \[ (7 + 10i) - 0 = 7 + 10i \]

Key Concepts

Distributive Law

Understanding the distributive law is crucial when working with complex numbers. The distributive law states that for any numbers or variables, the formula \(a(b+c) = ab + ac\) holds true. This law applies not only to real numbers but also to complex numbers, which consist of both real and imaginary parts.
When you come across an expression like \((8+9i)(2-i)\), you need to ensure that each part of the first term is multiplied by each part of the second term. Apply the distributive law as follows:

• Multiply \(8\) by \(2\) and then \(8\) by \(-i\) to get \(16 - 8i\).
• Multiply \(9i\) by \(2\) and then \(9i\) by \(-i\) to get \(18i - 9i^2\).
  

Remember that \(i^2 = -1\), so it simplifies to \(-9(-1)=9\). This transformation is pivotal for finding the combined real and imaginary parts.

Standard Form

Presenting complex numbers in a clear and consistent way is important for ease of understanding. The standard form for complex numbers is \(a + bi\), where \(a\) represents the real part, and \(b\) represents the imaginary part. For example, \(7 + 10i\) is in standard form.
To transform the result of the operation \((8+9i)(2-i)-(1-i)(1+i)\) to standard form, go through the following process:

• Perform multiplication and simplify the expression according to the distributive law.
• Combine like terms, ensuring all real parts (terms with no \(i\)) are added separately from the imaginary parts (terms with \(i\)).
  

This simplification yields the complex number in standard form, which is simple and ready for further operations or interpretation.

Real and Imaginary Parts

Complex numbers are composed of two parts: a real part and an imaginary part. For any complex number \(a + bi\):

• \(a\) refers to the real part.
• \(bi\) refers to the imaginary part.

In the expression \(7 + 10i\), \(7\) is the real part, and \(10i\) is the imaginary part.
Understanding how to separate these elements is vital. When you add or subtract complex numbers, you operate only within the respective parts. For instance:

• Real parts combine with real parts.
• Imaginary parts combine with imaginary parts.
  

This separation makes operations more manageable and helps in simplifying expressions accurately. Applying this approach helps ensure results are in the correct form to easily interpret their nature and magnitude.