Understanding Complex Numbers and Exponents

Oct 13, 2025 by windailys.com 44 views

The Expression $6 x i^3(-4 x i+5)$ : Simplifying Complex Numbers

Understanding Complex Numbers and Exponents

Complex numbers are mathematical expressions that combine real and imaginary parts. They are often represented in the form $a + bi$ , where $a$ is the real part and $b$ is the imaginary part. In this expression, $i$ is the imaginary unit, which is defined as the square root of $-1$ . When dealing with complex numbers, it's essential to understand the properties of exponents and how they interact with the imaginary unit.

Simplifying the Expression $6 x i^3(-4 x i+5)$

To simplify the given expression, we need to apply the rules of exponents and complex numbers. The expression $6 x i^3(-4 x i+5)$ can be rewritten as $6 x i^3 \cdot (-4 x i) + 6 x i^3 \cdot 5$ . Using the property of exponents that states $i^3 = -i$ , we can simplify the expression further.

Applying the Property of Exponents

The property of exponents states that $i^3 = -i$ . This means that when we raise $i$ to the power of $3$ , we get $-i$ . We can apply this property to the expression $6 x i^3(-4 x i+5)$ to simplify it.

Simplifying the Expression

Using the property of exponents, we can rewrite the expression as $6 x (-i) \cdot (-4 x i) + 6 x (-i) \cdot 5$ . Simplifying further, we get $24 x^2 i^2 + 30 x i$ . Now, we need to apply the property of exponents that states $i^2 = -1$ .

Applying the Property of Exponents

The property of exponents states that $i^2 = -1$ . This means that when we raise $i$ to the power of $2$ , we get $-1$ . We can apply this property to the expression $24 x^2 i^2 + 30 x i$ to simplify it.

Simplifying the Expression

Using the property of exponents, we can rewrite the expression as $24 x^2 \cdot (-1) + 30 x i$ . Simplifying further, we get $-24 x^2 + 30 x i$ . This is the simplified form of the expression $6 x i^3(-4 x i+5)$ .

Conclusion

Answer

The correct answer is b) $-24 x^2 + 30 x i$ .

Discussion

This problem requires a deep understanding of complex numbers and exponents. The student needs to apply the properties of exponents and complex numbers to simplify the given expression. The student also needs to be able to recognize the correct form of the simplified expression.

Tips and Tricks

Make sure to apply the properties of exponents and complex numbers correctly.
Use the correct form of the simplified expression.
Double-check your work to ensure that you have obtained the correct answer.

Common Mistakes

Failing to apply the properties of exponents and complex numbers correctly.
Using the incorrect form of the simplified expression.
Not double-checking the work to ensure that the correct answer has been obtained.

Real-World Applications

This problem has real-world applications in fields such as engineering, physics, and computer science. Complex numbers are used to represent electrical circuits, signal processing, and other applications. Understanding the properties of exponents and complex numbers is essential for solving problems in these fields.

Practice Problems

Simplify the expression $3 x i^2(-2 x i+3)$ .
Simplify the expression $2 x i^3(4 x i-5)$ .
Simplify the expression $5 x i^2(3 x i+2)$ .

Answer Key

a) $-6 x^2 + 9 x i$
b) $-8 x^2 + 15 x i$
c) $-15 x^2 + 10 x i$

Conclusion

In conclusion, the expression $6 x i^3(-4 x i+5)$ is equivalent to $-24 x^2 + 30 x i$ . This is the simplified form of the expression, which was obtained by applying the properties of exponents and complex numbers. The student needs to apply the properties of exponents and complex numbers correctly and use the correct form of the simplified expression to obtain the correct answer.
Q&A: Simplifying Complex Numbers

Q: What is the property of exponents that states $i^3 = -i$ ?

A: The property of exponents states that $i^3 = -i$ . This means that when we raise $i$ to the power of $3$ , we get $-i$ .

Q: How do we simplify the expression $6 x i^3(-4 x i+5)$ ?

A: To simplify the expression $6 x i^3(-4 x i+5)$ , we need to apply the rules of exponents and complex numbers. We can rewrite the expression as $6 x i^3 \cdot (-4 x i) + 6 x i^3 \cdot 5$ . Using the property of exponents that states $i^3 = -i$ , we can simplify the expression further.

Q: What is the simplified form of the expression $6 x i^3(-4 x i+5)$ ?

A: The simplified form of the expression $6 x i^3(-4 x i+5)$ is $-24 x^2 + 30 x i$ .

Q: How do we apply the property of exponents that states $i^2 = -1$ ?

A: To apply the property of exponents that states $i^2 = -1$ , we need to raise $i$ to the power of $2$ . This means that when we raise $i$ to the power of $2$ , we get $-1$ .

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

A: $i$ and $-i$ are two different imaginary units. $i$ is defined as the square root of $-1$ , while $-i$ is the negative of $i$ .

Q: How do we simplify the expression $3 x i^2(-2 x i+3)$ ?

A: To simplify the expression $3 x i^2(-2 x i+3)$ , we need to apply the rules of exponents and complex numbers. We can rewrite the expression as $3 x i^2 \cdot (-2 x i) + 3 x i^2 \cdot 3$ . Using the property of exponents that states $i^2 = -1$ , we can simplify the expression further.

Q: What is the simplified form of the expression $3 x i^2(-2 x i+3)$ ?

A: The simplified form of the expression $3 x i^2(-2 x i+3)$ is $-6 x^2 + 9 x i$ .

Q: How do we simplify the expression $2 x i^3(4 x i-5)$ ?

A: To simplify the expression $2 x i^3(4 x i-5)$ , we need to apply the rules of exponents and complex numbers. We can rewrite the expression as $2 x i^3 \cdot (4 x i) - 2 x i^3 \cdot 5$ . Using the property of exponents that states $i^3 = -i$ , we can simplify the expression further.

Q: What is the simplified form of the expression $2 x i^3(4 x i-5)$ ?

A: The simplified form of the expression $2 x i^3(4 x i-5)$ is $-8 x^2 + 15 x i$ .

Q: How do we simplify the expression $5 x i^2(3 x i+2)$ ?

A: To simplify the expression $5 x i^2(3 x i+2)$ , we need to apply the rules of exponents and complex numbers. We can rewrite the expression as $5 x i^2 \cdot (3 x i) + 5 x i^2 \cdot 2$ . Using the property of exponents that states $i^2 = -1$ , we can simplify the expression further.

Q: What is the simplified form of the expression $5 x i^2(3 x i+2)$ ?

A: The simplified form of the expression $5 x i^2(3 x i+2)$ is $-15 x^2 + 10 x i$ .

Conclusion

In conclusion, simplifying complex numbers requires a deep understanding of the properties of exponents and complex numbers. By applying the rules of exponents and complex numbers, we can simplify complex expressions and obtain the correct answer.