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Chapter 7: Problem 48

Multiply and simplify. Write each answer in the form \(a+b i\). $$-4 i(6-5 i)$$

Short Answer

Expert verified
-20 - 24i

Step by step solution

01

- Distribute the term

Distribute $$-4i$$ to each term within the parenthesis. This means multiplying $$-4i$$ with both $$6$$ and $$-5i$$.
02

- Multiply the first term

Multiply $$-4i$$ by $$6$$. This gives: $$-4i \times 6 = -24i.$$
03

- Multiply the second term

Multiply $$-4i$$ by $$-5i$$. This gives: $$-4i \times -5i = 20i^2.$$
04

- Simplify using \(i^2 = -1\)

Recall that $i^2$$ is equivalent to $$-1$$. So, replace $$i^2$$ in $$20i^2$$ with $$-1$$, giving: $$20 \times -1 = -20.$$
05

- Combine like terms

Combine the real and imaginary parts together. The real part is $$-20$$ and the imaginary part is $$-24i$$. So, the final answer in $$a+b i$$ form is: $$-20 - 24i.$$

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
The distributive property is a fundamental concept in algebra. It states that for any numbers (or terms) A, B, and C, the equation A(B + C) equals AB + AC. In essence, you are distributing the number outside the parentheses to each term inside the parentheses.
In our exercise, we applied this property to multiply the complex number -4i(6 - 5i). That means -4i was multiplied by 6 and also by -5i.
Imaginary Unit i
The imaginary unit is denoted by the letter '**i**' and is defined by the property i^2 = -1. This definition is crucial for understanding complex numbers, which are numbers that have both a real part and an imaginary part.
For example, in the given problem, we have imaginary units represented by -4i and -5i. The results from multiplying these will later involve i^2.
Simplifying Expressions
Simplifying expressions is an important skill in algebra. When working with complex numbers, this often involves using the properties of i.
In our example, after distributing -4i to both terms inside the parentheses:
  • Multiplying -4i by 6 gives -24i.
  • Multiplying -4i by -5i gives 20i^2.
We then use i^2 = -1 to simplify 20i^2 to -20. This crucial step helps us combine our results to form the final simplified expression.
Real and Imaginary Parts
Complex numbers have two components: the real part and the imaginary part. These are usually written in the form a + bi, where:**a** is the real part and **bi** is the imaginary part.
In our exercise, after simplifying, the expression -4i (6 - 5i) we end up with:
  • The real part: -20.
  • The imaginary part: -24i.
Combining these gives us the final answer in the required form a + bi: -20 - 24i.

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