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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))).
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