Simplify The Following Expression:${ 2i - (2 + 3i) + (1 - 8i) = }$

Mar 12, 2025 by ADMIN 68 views

Simplify the Expression: 2i - (2 + 3i) + (1 - 8i)

Introduction

In mathematics, simplifying complex expressions is a crucial skill that helps us solve problems efficiently. When dealing with expressions involving imaginary numbers, it's essential to understand the rules of arithmetic operations to simplify them correctly. In this article, we will simplify the given expression: $2i - (2 + 3i) + (1 - 8i)$ . We will break down the expression step by step, applying the rules of arithmetic operations to simplify it.

Understanding the Expression

The given expression is $2i - (2 + 3i) + (1 - 8i)$ . This expression involves imaginary numbers, which are denoted by the letter $i$ . The imaginary unit $i$ is defined as the square root of $-1$ , i.e., $i = \sqrt{-1}$ . In this expression, we have three terms: $2i$ , $-(2 + 3i)$ , and $1 - 8i$ .

Distributive Property

To simplify the expression, we will apply the distributive property, which states that for any real numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ . We will use this property to expand the terms in the expression.

Step 1: Expand the Negative Term

We will start by expanding the negative term $-(2 + 3i)$ . Using the distributive property, we can rewrite this term as $-2 - 3i$ .

Step 2: Expand the Positive Term

Next, we will expand the positive term $1 - 8i$ . Using the distributive property, we can rewrite this term as $1 - 8i$ .

Step 3: Combine Like Terms

Now that we have expanded the terms, we can combine like terms. The expression becomes $2i - 2 - 3i + 1 - 8i$ . We can combine the like terms $2i$ and $-3i$ to get $-i$ . Similarly, we can combine the like terms $-2$ and $1$ to get $-1$ .

Step 4: Simplify the Expression

After combining the like terms, the expression becomes $-i - 1 - 8i$ . We can combine the like terms $-i$ and $-8i$ to get $-9i$ . Therefore, the simplified expression is $-9i - 1$ .

Conclusion

In this article, we simplified the expression $2i - (2 + 3i) + (1 - 8i)$ step by step. We applied the distributive property to expand the terms and then combined like terms to simplify the expression. The final simplified expression is $-9i - 1$ .

Final Answer

The final answer is $\boxed{-9i - 1}$ .

Discussion

Simplifying complex expressions is an essential skill in mathematics. In this article, we demonstrated how to simplify the expression $2i - (2 + 3i) + (1 - 8i)$ using the distributive property and combining like terms. This skill is crucial in solving problems involving imaginary numbers and is an essential part of algebra and calculus.

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Imaginary Numbers" by Wolfram MathWorld

Introduction

In our previous article, we simplified the expression $2i - (2 + 3i) + (1 - 8i)$ step by step. We applied the distributive property to expand the terms and then combined like terms to simplify the expression. In this article, we will answer some frequently asked questions related to simplifying expressions with imaginary numbers.

Q&A

Q1: What is the distributive property?

A1: The distributive property is a mathematical rule that states that for any real numbers $a$ , $b$ , and $c$ , $a(b + c) = ab + ac$ . This property allows us to expand the terms in an expression by multiplying each term inside the parentheses by the term outside the parentheses.

Q2: How do I simplify an expression with imaginary numbers?

A2: To simplify an expression with imaginary numbers, you need to apply the distributive property to expand the terms and then combine like terms. You can also use the rules of arithmetic operations, such as addition and multiplication, to simplify the expression.

Q3: What is the difference between a real number and an imaginary number?

A3: A real number is a number that can be expressed as a decimal or a fraction, such as 3 or 1/2. An imaginary number, on the other hand, is a number that can be expressed as the square root of a negative number, such as $i = \sqrt{-1}$ .

Q4: How do I combine like terms in an expression?

A4: To combine like terms in an expression, you need to identify the terms that have the same variable or constant. You can then add or subtract these terms to simplify the expression.

Q5: What is the final simplified expression for $2i - (2 + 3i) + (1 - 8i)$ ?

A5: The final simplified expression for $2i - (2 + 3i) + (1 - 8i)$ is $-9i - 1$ .

Q6: Can I use the distributive property to simplify an expression with multiple terms?

A6: Yes, you can use the distributive property to simplify an expression with multiple terms. You can apply the distributive property to each term in the expression and then combine like terms to simplify the expression.

Q7: How do I check my work when simplifying an expression?

A7: To check your work when simplifying an expression, you can plug in some values for the variables and constants in the expression and see if the result is correct. You can also use a calculator or a computer program to check your work.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions with imaginary numbers. We discussed the distributive property, combining like terms, and checking work. We also provided some examples and exercises to help you practice simplifying expressions with imaginary numbers.

Final Answer

The final answer is $\boxed{-9i - 1}$ .

Discussion

Simplifying expressions with imaginary numbers is an essential skill in mathematics. In this article, we demonstrated how to simplify the expression $2i - (2 + 3i) + (1 - 8i)$ using the distributive property and combining like terms. This skill is crucial in solving problems involving imaginary numbers and is an essential part of algebra and calculus.

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Imaginary Numbers" by Wolfram MathWorld

Simplify The Following Expression:${ 2i - (2 + 3i) + (1 - 8i) = }$

Introduction

Understanding the Expression

Distributive Property

Step 1: Expand the Negative Term

Step 2: Expand the Positive Term

Step 3: Combine Like Terms

Step 4: Simplify the Expression

Conclusion

Final Answer

Discussion

Related Topics

References

Further Reading

Introduction

Q&A

Q1: What is the distributive property?

Q2: How do I simplify an expression with imaginary numbers?

Q3: What is the difference between a real number and an imaginary number?

Q4: How do I combine like terms in an expression?

Q5: What is the final simplified expression for $2i - (2 + 3i) + (1 - 8i)$ ?

Q6: Can I use the distributive property to simplify an expression with multiple terms?

Q7: How do I check my work when simplifying an expression?

Conclusion

Final Answer

Discussion

Related Topics

References

Further Reading

Introduction

Understanding the Expression

Distributive Property

Step 1: Expand the Negative Term

Step 2: Expand the Positive Term

Step 3: Combine Like Terms

Step 4: Simplify the Expression

Conclusion

Final Answer

Discussion

Related Topics

References

Further Reading

Introduction

Q&A

Q1: What is the distributive property?

Q2: How do I simplify an expression with imaginary numbers?

Q3: What is the difference between a real number and an imaginary number?

Q4: How do I combine like terms in an expression?

Q5: What is the final simplified expression for 2i−(2+3i)+(1−8i)2i - (2 + 3i) + (1 - 8i)2i−(2+3i)+(1−8i)?

Q6: Can I use the distributive property to simplify an expression with multiple terms?

Q7: How do I check my work when simplifying an expression?

Conclusion

Final Answer

Discussion

Related Topics

References

Further Reading

Q5: What is the final simplified expression for $2i - (2 + 3i) + (1 - 8i)$ ?