(a)
To show:
(a)
Answer to Problem 97E
Hence,
Explanation of Solution
Given:
Calculation:
Consider the following formula for positive integer's n:
To show
Recall the definition of a root of a
If
Start with
Clearly
For
For
For
Use of the additive law of exponents:
Thus,
Note that although
Here, each is the distinct 0th root, 1st root, and 2nd (square), and 3rd (cube) root respectively. This means that all the other 0th, 1st, 2nd (square), and 3rd (cube) roots are the same as
Since it seems that the formula works for lower powers of
That is, assume that
Check
This will make use of the additive law of exponents:
Thus,
Substitute
Because assumed
Hence,
Conclusion:
Hence,
(b)
To show: the
(b)
Answer to Problem 97E
Hence,
Explanation of Solution
Given:
Calculation:
Consider a complex number
To show that the
First recall the definition of a root of a complex number
If
This means that for any positive integer
To find the distinct
This means that:
These are
Use this result along with the given formula to find
For
For
In general, for
Which are all distinct.
Hence,
Conclusion:
Hence,
Chapter 8 Solutions
Precalculus: Mathematics for Calculus - 6th Edition
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- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning