Simplify The Expression $\frac{5}{4} - \left(\frac{7}{4} + 2i\right$\] And Write The Result In The Form $a + Bi$.A. $\frac{1}{2} - 2i$ B. $-\frac{1}{2} - 2i$ C. $-2 - \frac{1}{2}i$ D. $2 -

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. When dealing with complex numbers, it's essential to understand how to combine them using basic arithmetic operations. In this article, we will simplify the expression $\frac{5}{4} - \left(\frac{7}{4} + 2i\right)$ and write the result in the form $a + bi$. We will also explore the different options available and determine the correct answer.

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 $i^2 = -1$. Complex numbers have both a real part and an imaginary part, and they can be added, subtracted, multiplied, and divided just like real numbers.

Simplifying the Expression

Now, let's simplify the expression $\frac{5}{4} - \left(\frac{7}{4} + 2i\right)$. To do this, we need to follow the order of operations (PEMDAS):

1. Evaluate the expression inside the parentheses: $\frac{7}{4} + 2i$
2. Subtract the result from step 1 from $\frac{5}{4}$

Let's start by evaluating the expression inside the parentheses:

$\frac{7}{4} + 2i = \frac{7}{4} + \frac{8}{4}i = \frac{7 + 8i}{4}$

Now, let's subtract the result from $\frac{5}{4}$:

$\frac{5}{4} - \left(\frac{7}{4} + 2i\right) = \frac{5}{4} - \frac{7 + 8i}{4}$

To subtract these fractions, we need to have a common denominator, which is 4. So, we can rewrite the fractions as:

$\frac{5}{4} = \frac{5}{4}$

$\frac{7 + 8i}{4} = \frac{7}{4} + \frac{8}{4}i$

Now, we can subtract the fractions:

$\frac{5}{4} - \left(\frac{7}{4} + \frac{8}{4}i\right) = \frac{5}{4} - \frac{7}{4} - \frac{8}{4}i$

Combine like terms:

$\frac{5}{4} - \frac{7}{4} = \frac{5 - 7}{4} = -\frac{2}{4} = -\frac{1}{2}$

$-\frac{8}{4}i = -2i$

So, the simplified expression is:

$-\frac{1}{2} - 2i$

Conclusion

In this article, we simplified the expression $\frac{5}{4} - \left(\frac{7}{4} + 2i\right)$ and wrote the result in the form $a + bi$. We followed the order of operations and combined like terms to arrive at the final answer. The correct answer is $-\frac{1}{2} - 2i$.

Discussion

The discussion category for this article is mathematics. If you have any questions or comments about simplifying expressions or complex numbers, please feel free to share them below.

Final Answer

The final answer is $-\frac{1}{2} - 2i$.

Introduction

In our previous article, we simplified the expression $\frac{5}{4} - \left(\frac{7}{4} + 2i\right)$ and wrote the result in the form $a + bi$. We received many questions and comments from readers, and we're excited to address them in this Q&A article. If you have any questions or concerns about simplifying expressions or complex numbers, please feel free to ask.

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, 4, 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 simplify complex numbers?

A: To simplify complex numbers, you need to follow the order of operations (PEMDAS):

1. Evaluate the expression inside the parentheses.
2. Combine like terms.
3. Simplify the expression.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when you have multiple operations in an expression. The acronym PEMDAS stands for:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I add and subtract complex numbers?

A: To add and subtract complex numbers, you need to follow the same rules as adding and subtracting real numbers. However, when adding or subtracting complex numbers, you need to combine the real parts and the imaginary parts separately.

Q: What is the difference between $i$ and $-i$?

A: $i$ and $-i$ are two different imaginary units. $i$ is the imaginary unit that satisfies the equation $i^2 = -1$, while $-i$ is the negative of $i$. When working with complex numbers, you need to be careful to use the correct imaginary unit.

Q: Can you provide more examples of simplifying complex numbers?

A: Here are a few more examples of simplifying complex numbers:

• $\frac{3}{4} + \left(\frac{5}{4} - 2i\right) = \frac{3}{4} + \frac{5}{4} - \frac{8}{4}i = \frac{8}{4} - \frac{8}{4}i = 2 - 2i$
• $\frac{2}{3} - \left(\frac{4}{3} + 3i\right) = \frac{2}{3} - \frac{4}{3} - \frac{9}{3}i = -\frac{2}{3} - \frac{9}{3}i = -\frac{2}{3} - 3i$

Conclusion

In this Q&A article, we addressed many common questions and concerns about simplifying complex numbers. We hope that this article has been helpful in clarifying any confusion and providing more examples of simplifying complex numbers.

Final Answer

The final answer is $-\frac{1}{2} - 2i$.