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Chapter 7: Problem 46

Multiply and simplify. Write each answer in the form \(a+b i\). $$2 i(7+3 i)$$

Short Answer

Expert verified
-6 + 14i

Step by step solution

01

Distribute the imaginary unit

Use the distributive property to multiply the imaginary unit 2i with each term inside the parenthesis. That is, multiply 2i by 7 and then 2i by 3i: \[ 2i(7 + 3i) = 2i \times 7 + 2i \times 3i \]
02

Perform the multiplication

Multiply the constants and imaginary units: \[ 2i \times 7 = 14i \]Multiply the imaginary units, keeping in mind that \( i^2 = -1 \):\[ 2i \times 3i = 6i^2 = 6(-1) = -6 \]
03

Combine the terms

Add the real part and the imaginary part: \[ 14i - 6 = -6 + 14i \]

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Key Concepts

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

Distributive Property
When multiplying expressions, especially in algebra, the distributive property is a useful tool. It allows us to distribute or spread out one term across the terms within a parenthesis. For example, if you have the expression \(a(b+c)\), you can distribute \(a\) to both \(b\) and \(c\), giving us \(ab + ac\).

In our exercise, we use the distributive property to multiply \(2i(7+3i)\). Here, we distribute \(2i\) to both \(7\) and \(3i\), resulting in \(((2i \times 7) + (2i \times 3i))\). This step-by-step multiplication makes the process more manageable and easier to understand.
Imaginary Unit
The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined by the property that \(i^2 = -1\). This is quite unique because no real number has this property.

When multiplying complex numbers, it's crucial to remember how \(i\) behaves. For instance, in the exercise, when we multiply \((2i \times 3i)\), we get \(6i^2\). Since \(i^2 = -1\), this becomes \(6 \times (-1) = -6\). Knowing this helps simplify the results accurately.
Simplification
After performing the necessary multiplications, the next step is simplification. Simplification helps to express complex numbers in the standard form, \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.

For our given problem, we start with \(2i(7+3i) \), distribute and multiply to get \(14i - 6\). Notice that \(-6\) is the real part, and \(+14i\) is the imaginary part. Combining these terms, we get the simplified form: \(-6 + 14i\).

Simplification makes the results clear and standardized, ensuring we correctly represent complex numbers.

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