High School

7. [-/1 Points] DETAILS MCKTRIG8 8.3.033. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Use de Moivre's Theorem to find the following. Write your answer in standard form. (-2/3 – 21)4 Need Help? Read It 8. [-/1 Points] DETAILS MCKTRIG8 8.3.041. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Use de Moivre's Theorem to find the reciprocal of each number below. -V3 - 1 Need Help? Read It 11. [-/1 Points] DETAILS MCKTRIG8 8.3.051. MY NOTES ASK YOUR TEACHER PRACTICE ANOTHER Find the quotient 21/z2 in standard form. 21 = 373 + 3i, z2 = 21 21/22 = Write 21 and zz in trigonometric form and find their quotient again. 21/22 = Finally, convert the answer that is in trigonometric form to standard form. Are the two quotients equal? Yes O No

Answer :

Final answer:

The value of (-2/3 – 21)4 using de Moivre's Theorem is (445/9)^(2)(cos(4 * arctan(63/2)) + isin(4 * arctan(63/2))).

Explanation:

To find the value of (-2/3 – 21)4 using de Moivre's Theorem, we first need to convert the complex number to trigonometric form. The complex number (-2/3 – 21) can be written as z = r(cosθ + isinθ), where r is the magnitude of z and θ is the argument of z.

To find the magnitude of z, we can use the formula r = sqrt((-2/3)^2 + (-21)^2). Evaluating this expression gives us r = sqrt(4/9 + 441) = sqrt(445/9).

To find the argument of z, we can use the formula θ = arctan((-21)/(-2/3)). Evaluating this expression gives us θ = arctan(63/2).

Now that we have the magnitude and argument of z, we can raise z to the fourth power using de Moivre's Theorem. The formula for raising a complex number to a power is z^n = r^n(cos(nθ) + isin(nθ)). In this case, we have z^4 = (sqrt(445/9))^4(cos(4 * arctan(63/2)) + isin(4 * arctan(63/2))).

Finally, we can convert the result back to standard form. Evaluating the expression gives us z^4 = (445/9)^(4/2)(cos(4 * arctan(63/2)) + isin(4 * arctan(63/2))). Simplifying further gives us z^4 = (445/9)^(2)(cos(4 * arctan(63/2)) + isin(4 * arctan(63/2))).

Therefore, the value of (-2/3 – 21)4 using de Moivre's Theorem is (445/9)^(2)(cos(4 * arctan(63/2)) + isin(4 * arctan(63/2))).

Learn more about using de moivre's theorem to find the value of a complex number here:

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