Simplify \[$-5i \cdot 5\$\].Write Your Answer In The Form \[$a + Bi\$\].\[$\square\$\]
Introduction
In mathematics, simplifying complex expressions is a crucial skill that helps us solve problems efficiently. In this article, we will focus on simplifying the expression {-5i \cdot 5$}$, which involves multiplying a complex number by a real number. We will break down the process into manageable steps and provide a clear explanation of each step.
Understanding Complex Numbers
Before we dive into simplifying the expression, let's quickly review what complex numbers are. A complex number is a number 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$}$. In the expression {-5i \cdot 5$}$, we have a complex number {-5i$}$ being multiplied by a real number ${5\$}.
Step 1: Multiply the Complex Number by the Real Number
To simplify the expression {-5i \cdot 5$}$, we need to multiply the complex number {-5i$}$ by the real number ${5\$}. When multiplying a complex number by a real number, we can simply multiply the real part of the complex number by the real number, and multiply the imaginary part of the complex number by the real number.
-5i \cdot 5 = (-5 \cdot 5) + (-5 \cdot 5)i
Step 2: Simplify the Expression
Now that we have multiplied the complex number by the real number, we can simplify the expression by combining like terms.
(-5 \cdot 5) + (-5 \cdot 5)i = -25 + (-25)i
Step 3: Write the Answer in the Form {a + bi$}$
Finally, we can write the simplified expression in the form {a + bi$}$, where {a$}$ is the real part and {b$}$ is the imaginary part.
-25 + (-25)i = -25 - 25i
Conclusion
In this article, we simplified the expression {-5i \cdot 5$}$ by multiplying a complex number by a real number. We broke down the process into manageable steps and provided a clear explanation of each step. We also reviewed what complex numbers are and how to multiply them by real numbers. By following these steps, you can simplify complex expressions like {-5i \cdot 5$}$ and write the answer in the form {a + bi$}$.
Additional Examples
To reinforce your understanding of simplifying complex expressions, let's consider a few additional examples.
Example 1: Simplify ${3i \cdot 2\$}
3i \cdot 2 = (3 \cdot 2) + (3 \cdot 2)i
= 6 + 6i
Example 2: Simplify {-4i \cdot 3$}$
-4i \cdot 3 = (-4 \cdot 3) + (-4 \cdot 3)i
= -12 - 12i
Example 3: Simplify ${2i \cdot 4\$}
2i \cdot 4 = (2 \cdot 4) + (2 \cdot 4)i
= 8 + 8i
By practicing these examples, you can become more comfortable simplifying complex expressions and writing the answer in the form {a + bi$}$.
Final Thoughts
Q: What is the difference between a real number and a complex number?
A: A real number is a number that can be expressed without any imaginary part, such as ${3\$} or {-2$}$. A complex number, on the other hand, is a number that can be expressed in the form {a + bi$}$, where {a$}$ and {b$}$ are real numbers, and {i$}$ is the imaginary unit, which satisfies the equation {i^2 = -1$}$.
Q: How do I multiply a complex number by a real number?
A: To multiply a complex number by a real number, you can simply multiply the real part of the complex number by the real number, and multiply the imaginary part of the complex number by the real number.
Q: What is the formula for multiplying a complex number by a real number?
A: The formula for multiplying a complex number by a real number is:
(a + bi) \cdot c = (a \cdot c) + (b \cdot c)i
Q: How do I simplify a complex expression?
A: To simplify a complex expression, you can follow these steps:
- Multiply the complex number by the real number.
- Combine like terms.
- Write the answer in the form {a + bi$}$.
Q: What is the difference between {-5i \cdot 5$}$ and {-5i \cdot -5$}$?
A: The expression {-5i \cdot 5$}$ is equal to , while the expression {-5i \cdot -5$}$ is equal to . The difference between the two expressions is the sign of the real part.
Q: Can I simplify a complex expression with multiple complex numbers?
A: Yes, you can simplify a complex expression with multiple complex numbers by following the same steps as before. However, you may need to use the distributive property to multiply the complex numbers.
Q: How do I handle complex expressions with negative numbers?
A: When simplifying complex expressions with negative numbers, you can follow the same steps as before. However, you may need to use the properties of negative numbers, such as and .
Q: Can I use a calculator to simplify complex expressions?
A: Yes, you can use a calculator to simplify complex expressions. However, it's always a good idea to double-check your work by following the steps outlined in this article.
Q: What are some common mistakes to avoid when simplifying complex expressions?
A: Some common mistakes to avoid when simplifying complex expressions include:
- Forgetting to multiply the real part by the real number.
- Forgetting to multiply the imaginary part by the real number.
- Not combining like terms.
- Not writing the answer in the form {a + bi$}$.
By following these tips and avoiding common mistakes, you can become more confident in simplifying complex expressions and tackling more challenging problems.