Pregunta
- Find the square roots of the complex number
, leaving your answer in rectangular form
Ask by Gough Henry. in New Zealand
Jan 07,2025
Solución de inteligencia artificial de Upstudy
Respuesta verificada por el tutor
Responder
The square roots of
are
and
.
Solución
To find the square roots of the complex number
, we seek complex numbers
such that:
Expanding the left side:
Setting this equal to
, we obtain the system of equations:
From the second equation:
Substitute
into the first equation:
Multiply through by
:
Let
:
Solve for
:
Discard the negative solution since
must be non-negative:
Using
:
- If
, then . - If
, then .
Thus, the square roots of
are:
Respondido por UpStudy AI y revisado por un tutor profesional
Como
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Bonus Knowledge
To find the square roots of the complex number
, we can express it in polar form. First, calculate the modulus
using
, where
and
:
Next, find the argument
:
Thus, in polar form,
, where
is shorthand for
.
Now, to find the square roots, we use the formula for square roots in polar form:
So,
, and we have:
- For
:
Which gives us
.
- For
:
Leading to
.
Finally, after calculating the cosines and sines, you’ll find the roots in rectangular form:
And there you have it, the square roots of the complex number
!
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