Multiply.$(2+3i)^2$(2+3i)^2 = \square$(Type Your Answer In The Form $a + Bi$.)

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 complex numbers, specifically the expression $(2+3i)^2$. We will break down the process into manageable steps and provide a clear explanation of each step.

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 $a$, and the imaginary part is $bi$.

Multiplying Complex Numbers

To multiply complex numbers, we can use the distributive property, which states that for any complex numbers $a + bi$ and $c + di$, we have:

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

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

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

Multiplying $(2+3i)^2$

Now, let's apply the formula to multiply $(2+3i)^2$. We can start by expanding the expression:

$(2+3i)^2 = (2+3i)(2+3i)$

Using the distributive property, we get:

$(2+3i)(2+3i) = 2(2) + 2(3i) + 3i(2) + 3i(3i)$

Simplifying the expression, we get:

$(2+3i)(2+3i) = 4 + 6i + 6i + 9i^2$

Since $i^2 = -1$, we can substitute this value into the expression:

$(2+3i)(2+3i) = 4 + 12i - 9$

Combining like terms, we get:

$(2+3i)(2+3i) = -5 + 12i$

Conclusion

In this article, we have demonstrated how to multiply complex numbers using the distributive property. We have also applied this formula to multiply the expression $(2+3i)^2$. The result is $-5 + 12i$, which is a complex number in the form $a + bi$.

Tips and Tricks

• When multiplying complex numbers, make sure to use the distributive property to expand the expression.
• Simplify the expression by combining like terms.
• Use the fact that $i^2 = -1$ to simplify the expression further.

Practice Problems

• Multiply the complex numbers $(3+4i)(2+5i)$.
• Multiply the complex numbers $(1+2i)(3+4i)$.

Real-World Applications

Complex numbers have numerous applications in various fields, including:

• Algebra: Complex numbers are used to solve equations and inequalities.
• Geometry: Complex numbers are used to represent points and lines in the complex plane.
• Calculus: Complex numbers are used to represent functions and their derivatives.

Final Thoughts

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Introduction

In our previous article, we discussed how to multiply complex numbers using the distributive property. We also applied this formula to multiply the expression $(2+3i)^2$. In this article, we will provide a Q&A guide to help you understand the concept of multiplying complex numbers better.

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 apply the formula to multiply complex numbers?

A: To apply the formula, you need to follow these steps:

1. Expand the expression using the distributive property.
2. Simplify the expression by combining like terms.
3. Use the fact that $i^2 = -1$ to simplify the expression further.

Q: What is the difference between multiplying complex numbers and multiplying real numbers?

A: The main difference between multiplying complex numbers and multiplying real numbers is that complex numbers have an imaginary part, which is represented by $i$. When multiplying complex numbers, you need to take into account the imaginary part and use the distributive property to expand the expression.

Q: Can I multiply complex numbers with different powers of $i$?

A: Yes, you can multiply complex numbers with different powers of $i$. For example, you can multiply $(2+3i)$ with $(4+5i^2)$.

Q: How do I simplify complex numbers with different powers of $i$?

A: To simplify complex numbers with different powers of $i$, you need to use the fact that $i^2 = -1$. For example, if you have $(2+3i)(4+5i^2)$, you can simplify it by substituting $i^2 = -1$:

$(2+3i)(4+5i^2) = (2+3i)(4-5) = -2 + 12i$

Q: Can I multiply complex numbers with negative coefficients?

A: Yes, you can multiply complex numbers with negative coefficients. For example, you can multiply $(-2-3i)$ with $(4+5i)$.

Q: How do I apply the formula to multiply complex numbers with negative coefficients?

A: To apply the formula, you need to follow the same steps as before:

1. Expand the expression using the distributive property.
2. Simplify the expression by combining like terms.
3. Use the fact that $i^2 = -1$ to simplify the expression further.

For example, if you have $(-2-3i)(4+5i)$, you can simplify it by applying the formula:

$(-2-3i)(4+5i) = -8 - 10i - 12i - 15i^2$

Simplifying further, you get:

$(-2-3i)(4+5i) = -8 - 22i + 15 = 7 - 22i$

Q: Can I multiply complex numbers with complex coefficients?

A: Yes, you can multiply complex numbers with complex coefficients. For example, you can multiply $(2+3i)$ with $(4+5i)$.

Q: How do I apply the formula to multiply complex numbers with complex coefficients?

A: To apply the formula, you need to follow the same steps as before:

1. Expand the expression using the distributive property.
2. Simplify the expression by combining like terms.
3. Use the fact that $i^2 = -1$ to simplify the expression further.

For example, if you have $(2+3i)(4+5i)$, you can simplify it by applying the formula:

$(2+3i)(4+5i) = 8 + 10i + 12i + 15i^2$

Simplifying further, you get:

$(2+3i)(4+5i) = 8 + 22i - 15 = -7 + 22i$

Conclusion

In this article, we have provided a Q&A guide to help you understand the concept of multiplying complex numbers better. We have covered various topics, including the formula for multiplying complex numbers, applying the formula, and simplifying complex numbers with different powers of $i$. With practice, you will become proficient in multiplying complex numbers and be able to apply this skill to real-world problems.