Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
Related questions
Question
Evaluate the following square root expression:
![The expression shown in the image is:
\[
(-i)^{10} \sqrt{-324}
\]
### Explanation:
- **Complex Number \(i\):** In mathematics, \(i\) is the imaginary unit, which satisfies the equation \(i^2 = -1\).
- **Exponentiation of \(-i\):**
- The base is \(-i\), which is raised to the power of \(10\).
- To calculate \((-i)^{10}\), recognize that \((-i)^2 = -1\). Therefore, every even power of \(-i\) will be a power of \(-1\).
- Specifically, \((-i)^{10} = ((-i)^2)^5 = (-1)^5 = -1\).
- **Square Root of a Negative Number:**
- \(\sqrt{-324}\) involves taking the square root of a negative number, which involves the imaginary unit \(i\).
- This can be simplified to \(\sqrt{324} \cdot \sqrt{-1} = 18i\) because \(\sqrt{324} = 18\).
### Combining the Results:
Putting it all together, the original expression:
\[
(-i)^{10} \sqrt{-324}
\]
Simplifies to:
\[
-1 \cdot 18i = -18i
\]
Thus, the simplified result of the expression is \(-18i\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8ea37998-1c84-42d7-b5a7-416d3ab84958%2F382fbf50-f5b7-481b-bd1c-b16d273a13de%2F9x5xwja_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The expression shown in the image is:
\[
(-i)^{10} \sqrt{-324}
\]
### Explanation:
- **Complex Number \(i\):** In mathematics, \(i\) is the imaginary unit, which satisfies the equation \(i^2 = -1\).
- **Exponentiation of \(-i\):**
- The base is \(-i\), which is raised to the power of \(10\).
- To calculate \((-i)^{10}\), recognize that \((-i)^2 = -1\). Therefore, every even power of \(-i\) will be a power of \(-1\).
- Specifically, \((-i)^{10} = ((-i)^2)^5 = (-1)^5 = -1\).
- **Square Root of a Negative Number:**
- \(\sqrt{-324}\) involves taking the square root of a negative number, which involves the imaginary unit \(i\).
- This can be simplified to \(\sqrt{324} \cdot \sqrt{-1} = 18i\) because \(\sqrt{324} = 18\).
### Combining the Results:
Putting it all together, the original expression:
\[
(-i)^{10} \sqrt{-324}
\]
Simplifies to:
\[
-1 \cdot 18i = -18i
\]
Thus, the simplified result of the expression is \(-18i\).
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