Problem 55 - Intermediate Algebra

Question

Multiply. Write all answers in the form \(a+b i.\) $$ -5 i(5-5 i) $$

Step-by-Step Solution

Answer

The product is \(-25 - 25i\).

Step 1: Apply the Distributive Property

To multiply \(-5i(5-5i)\)we first need to distribute the \(-5i\) to each term within the parentheses. This results in two separate multiplications:1. \(-5i imes 5\)2. \(-5i imes -5i\)

Step 2: Multiply the Real and Imaginary Parts

Multiply \(-5i imes 5\):- The calculation is \((-5)(5)i = -25i\). Thus, the first term becomes \(-25i\).

Step 3: Multiply the Imaginary Parts

For \(-5i imes -5i\):- Multiply the coefficients \((-5)(-5) = 25\).- Multiply the imaginary units \(i imes i = i^2\).Since \(i^2 = -1\), we have:\(25i^2 = 25(-1) = -25\).So, the second term is \(-25\).

Step 4: Combine the Real and Imaginary Parts

Now, combine the terms from Steps 2 and 3. We combine the real part and the imaginary part to get the final answer: \(-25 + (-25i)\).

Key Concepts

Distributive Property

The distributive property is a fundamental concept in algebra which allows us to multiply a single term by each of the terms within a parenthesis. It essentially breaks down complex expressions into simpler parts, making calculations more manageable.
When you have an expression like \(-5i(5 - 5i)\), the distributive property tells us that you must multiply \(-5i\) by each term inside the parentheses.
This means performing two separate multiplications:

• First: \(-5i \times 5\)
• Second: \(-5i \times -5i\)

Each multiplication should be handled with care, respecting signs and factoring in both real and imaginary components.
This step-by-step approach simplifies what might first appear as a daunting calculation into smaller, more digestible pieces.

Imaginary Numbers

Imaginary numbers are central to understanding complex numbers and their operations. Imaginary numbers originate from the square root of negative one, denoted as \(i\).
The key rule to remember here is that \(i^2 = -1\).
This property plays a crucial role when multiplying or simplifying expressions involving \(i\). For example, in the multiplication of \(-5i \times -5i\):

• The coefficients \(-5 \times -5\) yield 25, as multiplying two negative numbers results in a positive number.
• Multiplying the imaginary units, \(i \times i\), produces \(i^2\), which simplifies to \(-1\).

This step reduces the term to a real number: \(-25\). Understanding this transformation is key to working with complex numbers and imaginary units effectively.

Combining Like Terms

Combining like terms is an essential skill when dealing with algebraic expressions.
It involves gathering terms that have the same variable component.
In complex numbers, it means merging real numbers with real numbers and imaginary numbers with imaginary numbers.
Consider the final step in our multiplication exercise: we have two resulting terms, \(-25i\) and \(-25\), from the previous operations.

• Here, \(-25\) is a real number (no imaginary unit \(i\) involved).
• Meanwhile, \(-25i\) includes the imaginary unit.

To combine these into a standard complex form \(a + bi\), simply place the real part together with the imaginary part:
\(-25 + (-25i)\).
This final expression is a tidy representation of the product in its simplest form.