Multiply And Simplify The Product: \[$-12i \times 3i\$\].Select The Product:A. 36 B. \[$-36\$\] C. \[$36i\$\] D. \[$-36i\$\]

Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including algebra, geometry, and calculus. In this article, we will focus on multiplying and simplifying complex numbers, specifically the product of two imaginary numbers. We will use the given problem, ${-12i \times 3i}$, to illustrate the steps involved in multiplying and simplifying complex numbers.

What are Complex Numbers?

Complex numbers are numbers that can be expressed in 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 denoted by ${a}$, and the imaginary part is denoted by ${bi}$.

Multiplying Complex Numbers

To multiply complex numbers, we can use the distributive property, which states that for any complex numbers ${a + bi}$ and ${c + di}$, the product is given by:

${(a + bi)(c + di) = ac + adi + bci + bdi^2}$

Since ${i^2 = -1}$, we can simplify the expression to:

${(a + bi)(c + di) = (ac - bd) + (ad + bc)i}$

Multiplying Two Imaginary Numbers

Now, let's apply this formula to the given problem, ${-12i \times 3i}$. We can rewrite the expression as:

${-12i \times 3i = (-12)(3)i^2}$

Using the fact that ${i^2 = -1}$, we can simplify the expression to:

${-12i \times 3i = (-12)(3)(-1)}$

Simplifying the Expression

Now, let's simplify the expression further:

${-12i \times 3i = 36}$

Conclusion

In conclusion, multiplying and simplifying complex numbers involves using the distributive property and the fact that ${i^2 = -1}$. By applying these rules, we can simplify the product of two imaginary numbers, ${-12i \times 3i}$, to ${36}$.

Answer

The correct answer is:

A. 36

Why is this the correct answer?

This is the correct answer because we have simplified the expression using the distributive property and the fact that ${i^2 = -1}$. The resulting expression is a real number, which is ${36}$.

What are the other options?

The other options are:

B. ${-36}$ C. ${36i}$ D. ${-36i}$

These options are incorrect because they do not result from simplifying the expression using the distributive property and the fact that ${i^2 = -1}$.

Why are these options incorrect?

These options are incorrect because they do not result from simplifying the expression using the distributive property and the fact that ${i^2 = -1}$. Specifically:

• Option B, ${-36}$, results from multiplying the real parts of the two complex numbers, but it does not take into account the imaginary parts.
• Option C, ${36i}$, results from multiplying the imaginary parts of the two complex numbers, but it does not take into account the real parts.
• Option D, ${-36i}$, results from multiplying the real parts of the two complex numbers and then multiplying the result by ${-1}$, but it does not take into account the imaginary parts.

Conclusion

In conclusion, multiplying and simplifying complex numbers involves using the distributive property and the fact that ${i^2 = -1}$. By applying these rules, we can simplify the product of two imaginary numbers, ${-12i \times 3i}$, to ${36}$. The correct answer is:

A. 36

Final Answer

The final answer is:

Introduction

In our previous article, we discussed how to multiply and simplify complex numbers, specifically the product of two imaginary numbers. In this article, we will provide a Q&A guide to help you understand the concepts and rules involved in multiplying and simplifying complex numbers.

Q: What is the formula for multiplying complex numbers?

A: The formula for multiplying complex numbers is:

${(a + bi)(c + di) = (ac - bd) + (ad + bc)i}$

Q: How do I simplify the expression when multiplying two imaginary numbers?

A: When multiplying two imaginary numbers, you can use the fact that ${i^2 = -1}$ to simplify the expression. For example, if you have ${-12i \times 3i}$, you can rewrite it as:

${-12i \times 3i = (-12)(3)i^2}$

Using the fact that ${i^2 = -1}$, you can simplify the expression to:

${-12i \times 3i = (-12)(3)(-1)}$

Q: What is the correct answer for the product of ${-12i \times 3i}$?

A: The correct answer is:

36

Q: Why is the correct answer 36?

A: The correct answer is 36 because we simplified the expression using the distributive property and the fact that ${i^2 = -1}$. The resulting expression is a real number, which is 36.

Q: What are the other options for the product of ${-12i \times 3i}$?

A: The other options are:

• ${-36}$
• ${36i}$
• ${-36i}$

Q: Why are these options incorrect?

A: These options are incorrect because they do not result from simplifying the expression using the distributive property and the fact that ${i^2 = -1}$. Specifically:

• Option B, ${-36}$, results from multiplying the real parts of the two complex numbers, but it does not take into account the imaginary parts.
• Option C, ${36i}$, results from multiplying the imaginary parts of the two complex numbers, but it does not take into account the real parts.
• Option D, ${-36i}$, results from multiplying the real parts of the two complex numbers and then multiplying the result by ${-1}$, but it does not take into account the imaginary parts.

Q: How do I know which option is correct?

A: To determine which option is correct, you need to simplify the expression using the distributive property and the fact that ${i^2 = -1}$. If the resulting expression is a real number, then that is the correct answer.

Q: What are some common mistakes to avoid when multiplying complex numbers?

A: Some common mistakes to avoid when multiplying complex numbers include:

• Not using the distributive property
• Not taking into account the imaginary parts
• Not using the fact that ${i^2 = -1}$
• Not simplifying the expression correctly

Conclusion

In conclusion, multiplying and simplifying complex numbers involves using the distributive property and the fact that ${i^2 = -1}$. By applying these rules, you can simplify the product of two imaginary numbers, ${-12i \times 3i}$, to 36. Remember to avoid common mistakes and always simplify the expression correctly.

Final Answer

The final answer is:

A. 36