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# Find all complex numbers \$z\$ such that \$z^4 = -4.\$ Note: All solutions should be expressed in the form \$a+bi\$, where \$a\$ and \$b\$ are real n

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Find all complex numbers such that

Note: All solutions should be expressed in the form , where a and b are real numbers.

May 18, 2021
edited by ch1ck3n  May 18, 2021

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a calculator can do this fast recommend mathway for this problem, but the answer is positive or negative the fourth root of positive 4 or negative 4.

May 18, 2021
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We can write -4 in exponential notation as 4e^(pi*i), so the equation is z^4 = 4e^(pi*i).

By Hamilton's Theorem, the solutions are z = 4^{1/4}*e^(pi*i/4), 4^{1/4}*e^(pi*i/4 + pi/4), 4^{1/4}*e^(pi*i/4 + 2*pi/4), and 4^{1/4}*e^(pi*i/4 + 3*pi/4).  Since 4^{1/4} = sqrt(2) and e^(pi*i/4) = (1 + i)/sqrt(2), the first solution is 1 + i.  Then the other roots work out as

4^{1/4}*e^(pi*i/4 + pi/4) = 1 - i,

4^{1/4}*e^(pi*i/4 + 2*pi/4) = -1 - i, and

4^{1/4}*e^(pi*i/4 + 3*pi/4) = -1 + i.

May 18, 2021