Explanation Given: T h e s i x t h r o o t s o f 729 Formula Used: Let the complex numberin polar form be given as z = r ( cos θ + i sin θ ) Then, for the integer n ≥ 2 , z has n distinct roots of the form w k = r n [ cos θ + 2 π k n + i sin θ + 2 π k n ] , k = 0 , 1 , 2..... n − 1 Calculation: Complex numberis given as follows: 729 729 + 0 i Converting the above complex number into polar form, we get r = ( 729 ) 2 + ( 0 ) 2 r = ( 729 ) 2 r = 729 And tan θ = 0 729 tan θ = 0 θ = tan − 1 ( 0 ) θ lies in first quadrant θ = 0 Thus, complex number in polar form is z = 729 ( cos 0 + i sin 0 ) Now, sixth roots of z = 729 ( cos 0 + i sin 0 ) are calculated as w k = 729 6 [ cos 0 + 2 π k 6 + i sin 0 + 2 π k 6 ] , k = 0 , 1 , 2 , 3 , 4 , 5 w k = ( 3 ) 6 6 [ cos π k 3 + i sin π k 3 ] , k = 0 , 1 , 2 , 3 , 4 , 5 w k = 3 [ cos π k 3 + i sin π k 3 ] , k = 0 , 1 , 2 , 3 , 4 , 5 Thus, we have w 0 = 3 [ cos π ( 0 ) 3 + i sin

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ISBN: 9781259976612

Author: Miller

Publisher: MCG CUSTOM

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Chapter 8.3, Problem 55PE

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To find the indicated complex roots by first writing the number in polar form and then write the result in rectangular form a+bi

The sixth roots of 729

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Chapter 8.3, Problem 54PE

Chapter 8.3, Problem 56PE

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To find the indicated complex roots by first writing the number in polar form and then write the result in rectangular form a + b i T h e s i x t h r o o t s o f 729

Solution Summary: The author explains the formula used to find the indicated complex roots by first writing the number in polar form and then write the result in rectangular form.

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To find the indicated complex roots by first writing the number in polar form and then write the result in rectangular form a+bi

The sixth roots of 729

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Chapter 8 Solutions

To find the indicated complex roots by first writing the number in polar form and then write the result in rectangular form a + b i T h e s i x t h r o o t s o f 729

Solution Summary: The author explains the formula used to find the indicated complex roots by first writing the number in polar form and then write the result in rectangular form.

Concept explainers

Videos

To find the indicated complex roots by first writing the number in polar form and then write the result in rectangular form a+bi The sixth roots of 729

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Chapter 8 Solutions

To find the indicated complex roots by first writing the number in polar form and then write the result in rectangular form a+bi

The sixth roots of 729