**Educational Example - Simplifying Expressions with Imaginary Numbers**In mathematics, particularly within the field of complex numbers, it is important to understand how to simplify expressions using the definition, identities, and properties of imaginary numbers. The task here involves matching terms in a list to their equivalent simplified forms.### Objective:Simplify each expression using the definition, identities, and properties of imaginary numbers. Match each term in the list on the left to its equivalent simplified form on the right.### Problems and Options for Matching:1. $(-\sqrt{7}i)^2 \cdot 2(i^5)^3 (i^9)^{-3}$2. $(3i (i^2)^(-1) (i^{5/2})^1)^2$3. $(i^8 i^5)^2 (i^7)^3$4. $(i^3 i^{-7} i^2)^3 (i^3 i^4)$### Potential Simplified Forms:- $1$- $-4$- $81$- $-i$### Instructions:1. Carefully simplify each expression using the properties of imaginary numbers $i$ (where $i^2 = -1$).2. Match each simplified term to one of the options provided.### Process and Explanation:Let’s simplify an example:**Example Explanation:**For the expression (1):1. $(-\sqrt{7}i)^2 \cdot 2(i^5)^3 (i^9)^{-3}$ - Simplify each part inside the parentheses. - Use $i^2 = -1$ identity and adjust exponents. Now, follow similar steps for the other expressions and values, and then match each simplified form correctly. Use the given options 1, -4, 81, and -i for the suitable simplified results.### Conclusion:Understanding the simplification process for imaginary numbers is crucial. Practice by simplifying each term and making the correct matches to reinforce this skill.### Additional Notes:If there are any questions or assistance needed, please use the "Ask for Help" option provided, or click "Turn it In" to submit your answers for evaluation.---This ends the lesson on simplifying expressions using imaginary numbers.

Simplify each expression using the definition, identities, and properties of imaginary numbers. Match each term in the list on the left to its equivalent simplified form on the right. 2 2. P[( PPA)( PPP)]2 4 -4 3. (PiS) ?( A) O 81 4. (P) ?(- P) 3(3 P) 4 3 NEXT QUESTION O ASK FOR HELP TURN IT IN

Simplify each expression using the definition, identities, and properties of imaginary numbers. Match each term in the list on the left to its equivalent simplified form on the right. 2 2. P[( PPA)( PPP)]2 4 -4 3. (PiS) ?( A) O 81 4. (P) ?(- P) 3(3 P) 4 3 NEXT QUESTION O ASK FOR HELP TURN IT IN

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Complex Numbers

Section: Chapter Questions

Problem 1T

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Question

**Educational Example - Simplifying Expressions with Imaginary Numbers**

In mathematics, particularly within the field of complex numbers, it is important to understand how to simplify expressions using the definition, identities, and properties of imaginary numbers. The task here involves matching terms in a list to their equivalent simplified forms.

### Objective:
Simplify each expression using the definition, identities, and properties of imaginary numbers. Match each term in the list on the left to its equivalent simplified form on the right.

### Problems and Options for Matching:
1. $(-\sqrt{7}i)^2 \cdot 2(i^5)^3 (i^9)^{-3}$
2. $(3i (i^2)^(-1) (i^{5/2})^1)^2$
3. $(i^8 i^5)^2 (i^7)^3$
4. $(i^3 i^{-7} i^2)^3 (i^3 i^4)$

### Potential Simplified Forms:
- $1$
- $-4$
- $81$
- $-i$

### Instructions:
1. Carefully simplify each expression using the properties of imaginary numbers $i$ (where $i^2 = -1$).
2. Match each simplified term to one of the options provided.

### Process and Explanation:
Let’s simplify an example:

**Example Explanation:**

For the expression (1):
1. $(-\sqrt{7}i)^2 \cdot 2(i^5)^3 (i^9)^{-3}$
- Simplify each part inside the parentheses.
- Use $i^2 = -1$ identity and adjust exponents.

Now, follow similar steps for the other expressions and values, and then match each simplified form correctly. Use the given options 1, -4, 81, and -i for the suitable simplified results.

### Conclusion:
Understanding the simplification process for imaginary numbers is crucial. Practice by simplifying each term and making the correct matches to reinforce this skill.

### Additional Notes:
If there are any questions or assistance needed, please use the "Ask for Help" option provided, or click "Turn it In" to submit your answers for evaluation.

---

This ends the lesson on simplifying expressions using imaginary numbers.

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