What Is The Value Of $5^3 I^9$?A. − 125 I -125 I − 125 I B. − 15 I -15 I − 15 I C. 15 D. 125 I 125 I 125 I

by ADMIN 110 views

Understanding the Problem

The problem involves evaluating the expression $5^3 i^9$, where ii is the imaginary unit with the property that i2=−1i^2 = -1. To find the value of this expression, we need to understand the properties of exponents and the behavior of the imaginary unit.

Properties of Exponents

When dealing with exponents, we need to remember the following rules:

  • am⋅an=am+na^m \cdot a^n = a^{m+n}
  • (am)n=am⋅n(a^m)^n = a^{m \cdot n}
  • am⋅bm=(ab)ma^m \cdot b^m = (ab)^m

Evaluating the Expression

To evaluate the expression $5^3 i^9$, we can start by simplifying the exponent of ii. Since i2=−1i^2 = -1, we can write i9i^9 as follows:

i9=(i2)4⋅i=(−1)4⋅i=1⋅i=ii^9 = (i^2)^4 \cdot i = (-1)^4 \cdot i = 1 \cdot i = i

Simplifying the Expression

Now that we have simplified the exponent of ii, we can rewrite the original expression as follows:

53i9=53i5^3 i^9 = 5^3 i

Evaluating the Exponent of 5

To evaluate the exponent of 5, we can simply multiply 5 by itself three times:

53=5⋅5⋅5=1255^3 = 5 \cdot 5 \cdot 5 = 125

Combining the Results

Now that we have evaluated the exponents of both 5 and ii, we can combine the results to find the final value of the expression:

53i9=125i5^3 i^9 = 125 i

Conclusion

In conclusion, the value of $5^3 i^9$ is 125i125 i. This result can be verified by using the properties of exponents and the behavior of the imaginary unit.

Comparison with Answer Choices

To compare our result with the answer choices, we can see that the correct answer is:

  • A. −125i-125 i (incorrect)
  • B. −15i-15 i (incorrect)
  • C. 15 (incorrect)
  • D. 125i125 i (correct)

Final Answer

The final answer is 125i\boxed{125 i}.

Frequently Asked Questions

Q: What is the value of $5^3 i^9$?

A: The value of $5^3 i^9$ is 125i125 i.

Q: Why is the value of $5^3 i^9$ equal to 125i125 i?

A: The value of $5^3 i^9$ is equal to 125i125 i because i9=ii^9 = i and 53=1255^3 = 125. Therefore, 53i9=125i5^3 i^9 = 125 i.

Q: What is the property of the imaginary unit ii that is used to simplify i9i^9?

A: The property of the imaginary unit ii that is used to simplify i9i^9 is i2=−1i^2 = -1. This allows us to write i9i^9 as (i2)4⋅i=(−1)4⋅i=1⋅i=i(i^2)^4 \cdot i = (-1)^4 \cdot i = 1 \cdot i = i.

Q: How do we evaluate the exponent of 5 in the expression 53i95^3 i^9?

A: We evaluate the exponent of 5 by multiplying 5 by itself three times: 53=5⋅5⋅5=1255^3 = 5 \cdot 5 \cdot 5 = 125.

Q: What is the final value of the expression 53i95^3 i^9?

A: The final value of the expression 53i95^3 i^9 is 125i125 i.

Q: How do we compare the result with the answer choices?

A: We compare the result with the answer choices by looking at the options provided. In this case, the correct answer is D. 125i125 i.

Q: What is the correct answer choice?

A: The correct answer choice is D. 125i125 i.

Q: What is the final answer?

A: The final answer is 125i\boxed{125 i}.

Additional Resources

If you have any further questions or need additional clarification on the topic, please refer to the following resources:

Conclusion

In conclusion, the value of $5^3 i^9$ is 125i125 i. This result can be verified by using the properties of exponents and the behavior of the imaginary unit. We hope this Q&A article has provided you with a better understanding of the topic. If you have any further questions, please don't hesitate to ask.