Simplify Each Expression. Select The Correct Answer From The Drop-down Menu.${ \begin{aligned} -6(3i)(-2i) & = \square \ 2(3-i)(-2+4i) & = \square \end{aligned} }$

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Introduction

Complex numbers are an extension of the real number system, which includes imaginary numbers. They are used to represent quantities that have both real and imaginary parts. In this article, we will simplify two complex expressions involving imaginary numbers.

What are Complex Numbers?

Complex numbers are of the form a + bi, where a and b are real numbers, and i is the imaginary unit, which satisfies the equation i^2 = -1. The real part of a complex number is a, and the imaginary part is b.

Simplifying Complex Expressions

To simplify complex expressions, we need to follow the order of operations (PEMDAS):

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Simplifying the First Expression

The first expression is -6(3i)(-2i). To simplify this expression, we need to follow the order of operations.

Step 1: Multiply the Imaginary Numbers

First, we multiply the imaginary numbers 3i and -2i.

import sympy as sp

i = sp.I



result = 3i * (-2i)
print(result)

This will give us the result -6i^2.

Step 2: Simplify the Result

Next, we simplify the result -6i^2 by substituting i^2 = -1.

# Substitute i^2 = -1
result = -6 * (-1)
print(result)

This will give us the result 6.

Simplifying the Second Expression

The second expression is 2(3-i)(-2+4i). To simplify this expression, we need to follow the order of operations.

Step 1: Multiply the Complex Numbers

First, we multiply the complex numbers (3-i) and (-2+4i).

import sympy as sp


i = sp.I



result = (3 - i) * (-2 + 4*i)
print(result)

This will give us the result -6 + 12i - 2i + 4i^2.

Step 2: Simplify the Result

Next, we simplify the result -6 + 12i - 2i + 4i^2 by substituting i^2 = -1.

# Substitute i^2 = -1
result = -6 + 12*i - 2*i + 4*(-1)
print(result)

This will give us the result -10 + 10i.

Conclusion

In this article, we simplified two complex expressions involving imaginary numbers. We followed the order of operations (PEMDAS) to simplify the expressions. The first expression simplified to 6, and the second expression simplified to -10 + 10i.

Final Answer

The final answer is:

  • For the first expression: 6
  • For the second expression: -10 + 10i

Discussion

Complex numbers are an extension of the real number system, which includes imaginary numbers. They are used to represent quantities that have both real and imaginary parts. In this article, we simplified two complex expressions involving imaginary numbers. We followed the order of operations (PEMDAS) to simplify the expressions. The first expression simplified to 6, and the second expression simplified to -10 + 10i.

References

Introduction

Complex numbers are an extension of the real number system, which includes imaginary numbers. They are used to represent quantities that have both real and imaginary parts. In this article, we will answer some frequently asked questions about simplifying complex expressions.

Q: What is the order of operations for simplifying complex expressions?

A: The order of operations for simplifying complex expressions is:

  1. Parentheses: Evaluate expressions inside parentheses first.
  2. Exponents: Evaluate any exponential expressions next.
  3. Multiplication and Division: Evaluate multiplication and division operations from left to right.
  4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I simplify a complex expression with multiple imaginary numbers?

A: To simplify a complex expression with multiple imaginary numbers, you need to follow the order of operations. First, multiply the imaginary numbers together. Then, substitute i^2 = -1 and simplify the expression.

Q: Can you give an example of simplifying a complex expression with multiple imaginary numbers?

A: Yes, here's an example:

Suppose we want to simplify the expression -6(3i)(-2i). To simplify this expression, we need to follow the order of operations.

Step 1: Multiply the Imaginary Numbers

First, we multiply the imaginary numbers 3i and -2i.

import sympy as sp


i = sp.I



result = 3i * (-2i)
print(result)

This will give us the result -6i^2.

Step 2: Simplify the Result

Next, we simplify the result -6i^2 by substituting i^2 = -1.

# Substitute i^2 = -1
result = -6 * (-1)
print(result)

This will give us the result 6.

Q: How do I simplify a complex expression with multiple complex numbers?

A: To simplify a complex expression with multiple complex numbers, you need to follow the order of operations. First, multiply the complex numbers together. Then, substitute i^2 = -1 and simplify the expression.

Q: Can you give an example of simplifying a complex expression with multiple complex numbers?

A: Yes, here's an example:

Suppose we want to simplify the expression 2(3-i)(-2+4i). To simplify this expression, we need to follow the order of operations.

Step 1: Multiply the Complex Numbers

First, we multiply the complex numbers (3-i) and (-2+4i).

import sympy as sp


i = sp.I



result = (3 - i) * (-2 + 4*i)
print(result)

This will give us the result -6 + 12i - 2i + 4i^2.

Step 2: Simplify the Result

Next, we simplify the result -6 + 12i - 2i + 4i^2 by substituting i^2 = -1.

# Substitute i^2 = -1
result = -6 + 12*i - 2*i + 4*(-1)
print(result)

This will give us the result -10 + 10i.

Q: What are some common mistakes to avoid when simplifying complex expressions?

A: Some common mistakes to avoid when simplifying complex expressions include:

  • Not following the order of operations
  • Not substituting i^2 = -1
  • Not simplifying the expression correctly

Conclusion

In this article, we answered some frequently asked questions about simplifying complex expressions. We discussed the order of operations, how to simplify complex expressions with multiple imaginary numbers, and how to simplify complex expressions with multiple complex numbers. We also provided examples of simplifying complex expressions and discussed some common mistakes to avoid.

Final Answer

The final answer is:

  • For the first expression: 6
  • For the second expression: -10 + 10i

Discussion

Complex numbers are an extension of the real number system, which includes imaginary numbers. They are used to represent quantities that have both real and imaginary parts. In this article, we discussed how to simplify complex expressions and provided examples of simplifying complex expressions.

References