Problem 58 Multiply. Give answers in standa... [FREE SOLUTION]

Chapter 10: Problem 58

Multiply. Give answers in standard form. $$ 3 i(4+9 i) $$

Short Answer

Expert verified

-27 + 12i

Step by step solution

- Distribute the terms

Use the distributive property to multiply each term inside the parentheses by the term outside. Multiply 3i by each term inside the parentheses: 3i(4) + 3i(9i)

- Perform the multiplications

Calculate the multiplications separately: 3i * 4 = 12i 3i * 9i = 27i^2

- Simplify the imaginary term

Remember that i^2 = -1So, convert 27i^2 to -27: 27i^2 = 27(-1) = -27

- Combine the results

Add the results together: 12i - 27

- Write in standard form

Standard form for a complex number is a + bi. Therefore, the final answer is: -27 + 12i

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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 algebraic property used to simplify expressions and solve equations. It states that multiplying a sum by a number is the same as multiplying each addend individually and then adding the results. For example, in the expression $3i(4 + 9i)$, the distributive property tells us to multiply $3i$ by both $4$ and $9i$ separately. This results in:
$3i \times 4$ and $3i \times 9i$.
After applying the distributive property, combining like terms becomes easier, leading to further simplification of complex expressions.

Imaginary Unit

The imaginary unit, denoted as $i$, is a mathematical concept used to extend the real number system. It is defined by the property that $i^2 = -1$. This helps in simplifying expressions involving square roots of negative numbers.
For example, when multiplying two complex numbers, like $3i \times 9i$, we use the fact that $i^2 = -1$. So, $27i^2$ simplifies to $27 \times -1$, which equals $-27$.
This property allows us to manage and manipulate complex numbers smoothly, especially when they involve imaginary units.

Standard Form of a Complex Number

A complex number is usually written in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit. This is known as the standard form.
The number $a$ is the real part, and $b$ is the imaginary part. For example, after simplifying $3i(4 + 9i)$, we reach the form $-27 + 12i$. Here:

The real part is $-27$
The imaginary part is $12i$

Writing complex numbers in this form makes them easier to add, subtract, and compare. It also helps in visualizing them on the complex plane, where the x-axis represents the real part and the y-axis signifies the imaginary part.

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91影视

Multiply. Give answers in standard form. $$ 3 i(4+9 i) $$

Short Answer

Step by step solution

- Distribute the terms

- Perform the multiplications

- Simplify the imaginary term

- Combine the results

- Write in standard form

Key Concepts

Distributive Property

Imaginary Unit

Standard Form of a Complex Number

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