Simplify. Write Your Answer In The Form Of $a + Bi$, Where $a$ And $b$ Are Real Numbers.$(8+\sqrt{-25})(6+\sqrt{-64})$A. 182 B. \$1,648+662 I$[/tex\] C. $182 I$ D. $8+94 I$

Introduction

Complex numbers are an essential part of mathematics, and they have numerous applications in various fields, including engineering, physics, and computer science. In this article, we will focus on simplifying complex expressions, specifically the product of two complex numbers in the form of $(a + bi)$, where $a$ and $b$ are real numbers.

Understanding Complex Numbers

Before we dive into simplifying complex expressions, let's briefly review the concept of complex numbers. A complex number is a number that can be expressed in the form of $a + bi$, where $a$ is the real part and $b$ is the imaginary part. The imaginary part is denoted by $i$, which is the square root of $-1$. Complex numbers can be represented graphically on a complex plane, with the real part on the x-axis and the imaginary part on the y-axis.

Simplifying Complex Expressions

To simplify complex expressions, we need to follow the order of operations (PEMDAS):

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

Simplifying the Given Expression

Now, let's simplify the given expression: $(8+\sqrt{-25})(6+\sqrt{-64})$. To simplify this expression, we need to follow the order of operations.

Step 1: Simplify the Square Roots

First, let's simplify the square roots:

$\sqrt{-25} = \sqrt{-1 \cdot 25} = 5i$

$\sqrt{-64} = \sqrt{-1 \cdot 64} = 8i$

Step 2: Substitute the Simplified Square Roots

Now, let's substitute the simplified square roots back into the original expression:

$(8+5i)(6+8i)$

Step 3: Multiply the Complex Numbers

To multiply complex numbers, we need to follow the distributive property:

$(8+5i)(6+8i) = 8(6+8i) + 5i(6+8i)$

$= 48 + 64i + 30i + 40i^2$

Step 4: Simplify the Expression

Now, let's simplify the expression:

$= 48 + 94i + 40i^2$

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

$= 48 + 94i + 40(-1)$

$= 48 + 94i - 40$

$= 8 + 94i$

Conclusion

In this article, we simplified the complex expression $(8+\sqrt{-25})(6+\sqrt{-64})$ using the order of operations and the distributive property. We found that the simplified expression is $8 + 94i$. This result is consistent with option D.

Discussion

Complex numbers are an essential part of mathematics, and they have numerous applications in various fields. Simplifying complex expressions is a crucial skill that can be used to solve problems in mathematics, physics, and engineering. In this article, we provided a step-by-step guide on how to simplify complex expressions, specifically the product of two complex numbers in the form of $(a + bi)$, where $a$ and $b$ are real numbers.

Frequently Asked Questions

• What is the order of operations? The order of operations is a set of rules that tells us which operations to perform first when we have multiple operations in an expression. The order of operations is: Parentheses, Exponents, Multiplication and Division, and Addition and Subtraction.
• How do I simplify complex expressions? To simplify complex expressions, we need to follow the order of operations and the distributive property. We need to simplify the square roots, substitute the simplified square roots back into the original expression, multiply the complex numbers, and simplify the expression.
• What is the distributive property? The distributive property is a rule that tells us how to multiply complex numbers. It states that $(a + bi)(c + di) = ac + adi + bci + bdi^2$.

References

• "Complex Numbers" by Khan Academy
• "Simplifying Complex Expressions" by Mathway
• "Complex Numbers and the Distributive Property" by Purplemath
  

Introduction

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields, including engineering, physics, and computer science. In our previous article, we provided a step-by-step guide on how to simplify complex expressions, specifically the product of two complex numbers in the form of $(a + bi)$, where $a$ and $b$ are real numbers. In this article, we will answer some frequently asked questions about complex numbers and provide additional information on simplifying expressions.

Q&A: Complex Numbers

Q: What is the difference between a real number and a complex number?

A: A real number is a number that can be expressed in the form of $a$, where $a$ is a real number. A complex number, on the other hand, is a number that can be expressed in the form of $a + bi$, where $a$ and $b$ are real numbers and $i$ is the square root of $-1$.

Q: How do I simplify complex expressions?

A: To simplify complex expressions, you need to follow the order of operations and the distributive property. You need to simplify the square roots, substitute the simplified square roots back into the original expression, multiply the complex numbers, and simplify the expression.

Q: What is the distributive property?

A: The distributive property is a rule that tells us how to multiply complex numbers. It states that $(a + bi)(c + di) = ac + adi + bci + bdi^2$.

Q: How do I multiply complex numbers?

A: To multiply complex numbers, you need to follow the distributive property. You need to multiply the real parts and the imaginary parts separately, and then combine the results.

Q: What is the conjugate of a complex number?

A: The conjugate of a complex number is a complex number that has the same real part and the opposite imaginary part. For example, the conjugate of $a + bi$ is $a - bi$.

Q: How do I add and subtract complex numbers?

A: To add and subtract complex numbers, you need to add or subtract the real parts and the imaginary parts separately.

Q: What is the modulus of a complex number?

A: The modulus of a complex number is the distance from the origin to the point representing the complex number on the complex plane. It is calculated using the formula $|a + bi| = \sqrt{a^2 + b^2}$.

Q: How do I divide complex numbers?

A: To divide complex numbers, you need to multiply the numerator and the denominator by the conjugate of the denominator.

Simplifying Complex Expressions: Additional Examples

Example 1: Simplify the expression $(3 + 4i)(2 - 5i)$

To simplify this expression, we need to follow the distributive property:

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

$= 6 - 15i + 8i - 20i^2$

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

$= 6 - 15i + 8i + 20$

$= 26 - 7i$

Example 2: Simplify the expression $(5 - 3i)(3 + 2i)$

To simplify this expression, we need to follow the distributive property:

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

$= 15 + 10i - 9i - 6i^2$

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

$= 15 + 10i - 9i + 6$

$= 21 + i$

Conclusion

In this article, we answered some frequently asked questions about complex numbers and provided additional information on simplifying expressions. We also provided two additional examples of simplifying complex expressions using the distributive property. We hope that this article has been helpful in understanding complex numbers and simplifying expressions.

Discussion

Complex numbers are a fundamental concept in mathematics, and they have numerous applications in various fields. Simplifying complex expressions is a crucial skill that can be used to solve problems in mathematics, physics, and engineering. In this article, we provided a step-by-step guide on how to simplify complex expressions, specifically the product of two complex numbers in the form of $(a + bi)$, where $a$ and $b$ are real numbers.

Frequently Asked Questions

• What is the difference between a real number and a complex number?
• How do I simplify complex expressions?
• What is the distributive property?
• How do I multiply complex numbers?
• What is the conjugate of a complex number?
• How do I add and subtract complex numbers?
• What is the modulus of a complex number?
• How do I divide complex numbers?

References

• "Complex Numbers" by Khan Academy
• "Simplifying Complex Expressions" by Mathway
• "Complex Numbers and the Distributive Property" by Purplemath