Simplify The Expression { (10+3i)(10-3i)$}$.A. 91 B. 109 C. ${ 100-30i\$} D. ${ 100+30i\$}
Introduction
In this article, we will simplify the expression {(10+3i)(10-3i)$}$. This involves using the distributive property and the concept of complex conjugates to arrive at the final result. We will break down the solution into manageable steps, making it easy to follow and understand.
Understanding Complex Conjugates
Before we dive into the solution, let's briefly discuss complex conjugates. A complex conjugate is a complex number that has the same real part but an opposite imaginary part. In other words, if we have a complex number in the form {a+bi$}$, its conjugate is {a-bi$}$. Complex conjugates are important in mathematics, particularly in algebra and calculus.
Step 1: Apply the Distributive Property
To simplify the expression {(10+3i)(10-3i)$}$, we will apply the distributive property. This property states that for any real numbers {a$}$, {b$}$, and {c$}$, we have:
{a(b+c) = ab + ac$}$
Using this property, we can expand the expression as follows:
{(10+3i)(10-3i) = 10(10-3i) + 3i(10-3i)$}$
Step 2: Simplify the Expression
Now, let's simplify the expression further. We can start by multiplying the terms inside the parentheses:
${10(10-3i) = 100 - 30i\$}
${3i(10-3i) = 30i - 9i^2\$}
Since {i^2 = -1$}$, we can substitute this value into the expression:
${30i - 9i^2 = 30i - 9(-1) = 30i + 9\$}
Now, let's combine the two simplified expressions:
${100 - 30i + 30i + 9 = 100 + 9\$}
Step 3: Final Result
After simplifying the expression, we arrive at the final result:
{(10+3i)(10-3i) = 109$}$
Conclusion
In this article, we simplified the expression {(10+3i)(10-3i)$}$ using the distributive property and the concept of complex conjugates. We broke down the solution into manageable steps, making it easy to follow and understand. The final result is ${109\$}, which is the correct answer.
Comparison of Options
Let's compare our final result with the options provided:
A. 91 B. 109 C. ${100-30i\$} D. ${100+30i\$}
Our final result, ${109\$}, matches option B. Therefore, the correct answer is B.
Key Takeaways
- Complex conjugates are important in mathematics, particularly in algebra and calculus.
- The distributive property can be used to simplify complex expressions.
- By breaking down the solution into manageable steps, we can arrive at the final result more easily.
Frequently Asked Questions
Q: What is a complex conjugate? A: A complex conjugate is a complex number that has the same real part but an opposite imaginary part.
Q: How do I apply the distributive property? A: To apply the distributive property, multiply the terms inside the parentheses and then combine like terms.
Introduction
In our previous article, we simplified the expression {(10+3i)(10-3i)$}$ using the distributive property and the concept of complex conjugates. We arrived at the final result of ${109\$}. In this article, we will provide a Q&A guide to help you better understand the solution and address any questions you may have.
Q&A Guide
Q: What is a complex number? A: A complex number is a number that can be expressed in the form {a+bi$}$, where {a$}$ is the real part and {b$}$ is the imaginary part.
Q: What is a complex conjugate? A: A complex conjugate is a complex number that has the same real part but an opposite imaginary part. In other words, if we have a complex number in the form {a+bi$}$, its conjugate is {a-bi$}$.
Q: How do I apply the distributive property? A: To apply the distributive property, multiply the terms inside the parentheses and then combine like terms. For example, if we have the expression {(a+b)(c+d)$}$, we can expand it as follows:
{(a+b)(c+d) = ac + ad + bc + bd$}$
Q: What is the difference between the distributive property and the commutative property? A: The distributive property states that for any real numbers {a$}$, {b$}$, and {c$}$, we have:
{a(b+c) = ab + ac$}$
The commutative property states that for any real numbers {a$}$ and {b$}$, we have:
{a+b = b+a$}$
Q: How do I simplify complex expressions? A: To simplify complex expressions, you can use the following steps:
- Apply the distributive property to expand the expression.
- Combine like terms.
- Simplify any complex numbers or expressions.
Q: What is the final result of the expression {(10+3i)(10-3i)$}$? A: The final result is ${109\$}.
**Q: Why is the final result $109\$} and not ${100-30i\$} or ${100+30i\$}?** A$ because when we simplify the expression, we get:
{(10+3i)(10-3i) = 100 - 30i + 30i + 9 = 109$}$
The other options, ${100-30i\$} and ${100+30i\$}, are not correct because they do not take into account the imaginary part of the expression.
Q: Can I use the distributive property to simplify other complex expressions? A: Yes, you can use the distributive property to simplify other complex expressions. Just remember to apply the distributive property to each term inside the parentheses and then combine like terms.
Conclusion
In this article, we provided a Q&A guide to help you better understand the solution to the expression {(10+3i)(10-3i)$}$. We addressed common questions and provided examples to help you apply the distributive property and simplify complex expressions.
Key Takeaways
- Complex conjugates are important in mathematics, particularly in algebra and calculus.
- The distributive property can be used to simplify complex expressions.
- By breaking down the solution into manageable steps, we can arrive at the final result more easily.
Frequently Asked Questions
Q: What is a complex number? A: A complex number is a number that can be expressed in the form {a+bi$}$, where {a$}$ is the real part and {b$}$ is the imaginary part.
Q: What is a complex conjugate? A: A complex conjugate is a complex number that has the same real part but an opposite imaginary part.
Q: How do I apply the distributive property? A: To apply the distributive property, multiply the terms inside the parentheses and then combine like terms.
Q: What is the final result of the expression {(10+3i)(10-3i)$}$? A: The final result is ${109\$}.
Additional Resources
If you want to learn more about complex numbers and the distributive property, here are some additional resources:
- Khan Academy: Complex Numbers
- Mathway: Distributive Property
- Wolfram Alpha: Complex Numbers
We hope this Q&A guide has been helpful in understanding the solution to the expression {(10+3i)(10-3i)$}$. If you have any further questions or need additional help, please don't hesitate to ask.