Problem 42 Multiply and simplify. $$-4 i(... [FREE SOLUTION]

Chapter 11: Problem 42

Multiply and simplify. $$-4 i(6+11 i)$$

Short Answer

Expert verified

The short answer is $44 - 24i$.

Step by step solution

Identify real and imaginary parts

First, we need to recognize the real and imaginary parts of the complex numbers. In this case, the complex number is $6 + 11i$, and the real number is $-4i$.

Perform the multiplication

Next, we'll multiply $-4i$ with the complex number. We'll distribute the $-4i$ to the real and imaginary parts separately. \[ -4i (6 + 11i) = (-4i \times 6) + (-4i \times 11i) \]

Simplify the expression

Finally, we'll simplify the expression by performing the operations within the parentheses. \[ (-4i \times 6) + (-4i \times 11i) = -24i - 44i^2 \] Since $i^2 = -1$, we can replace $i^2$ with $-1$ and further simplify the expression. \[ -24i - 44i^2 = -24i - 44(-1) = -24i + 44 \] The simplified expression is $44 - 24i$.

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Key Concepts

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

Real and Imaginary Parts

Complex numbers are made up of two parts: a real part and an imaginary part. The real part is a regular number, like 6 in the expression $6 + 11i$. The imaginary part involves $i$, which stands for the square root of -1. In our example, 11 accompanies the imaginary unit, so the imaginary part is 11i. Knowing how to identify these components is crucial because each part behaves differently when we perform operations, like multiplication, on the complex numbers. Identifying these parts before proceeding with multiplication allows us to handle each component appropriately and ensures that our calculations are accurate.

Distributed Multiplication

In algebra, distribution is a handy tool that helps us multiply a single term with multiple terms inside a parenthesis. When working with complex numbers, like multiplying $-4i$ with $6 + 11i$, this is an essential step. Here鈥檚 how it works:

First, distribute $-4i$ to the real part: $-4i \times 6$.
Next, distribute $-4i$ to the imaginary part: $-4i \times 11i$.

By tackling each component separately, we've efficiently broken down the multiplication task into smaller, manageable steps. This approach makes it easier to handle the complex multiplication without missing any detail.

Simplification of Complex Expressions

Once we have distributed the multiplication across the components, the next step is to simplify the resulting expression. Simplification often involves combining like terms and using known values, such as $i^2 = -1$.

For example, in our expression from the multiplication, we have
$(-4i \times 6) + (-4i \times 11i) = -24i - 44i^2$.
We know that $i^2$ is equal to -1, so $-44i^2$ becomes $-44(-1)$, which simplifies to 44.

Replace $i^2$ with -1.
Combine the resulting terms if possible: here, it becomes $-24i + 44$.

After these steps, we have a much simpler expression $44 - 24i$. Such clear simplification not only makes the expression easier to understand but also ensures the results are ready for further mathematical operations if needed.

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91影视

Multiply and simplify. $$-4 i(6+11 i)$$

Short Answer

Step by step solution

Identify real and imaginary parts

Perform the multiplication

Simplify the expression

Key Concepts

Real and Imaginary Parts

Distributed Multiplication

Simplification of Complex Expressions

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