>> In this example, we want to find the square root of the complex number, negative 3 plus 2i.
To begin, we'll z equal the complex number, negative 3 plus 2i.
So this would be equal to the square root of z. We're using rational exponents.
This would be equal to z to the one half power.
Now, for the first step, we'll write the complex number, negative 3 plus 2i, in polar form,
also referred to as trigonometric form where z is equal
to r times the quantity cosine theta plus i sine theta.
And we have our formulas here for r and theta.
So once we write z in polar form, we'll apply Dumas' theorem to determine z raised
to the power of n. And in our case, n will be equal to one half.
So we'll have z to the one half equals r
to the one half times the quantity cosine one half theta plus i sine one half theta.
Let's begin by plotting the complex number on the complex coordinate plane.
Notice how in this form, x would be negative 3 and y would be positive 2
where the x axis is the real axis and the y axis is the imaginary axis.
So we'd go left 3 units and up 2 units.
So this point here would represent the complex number on the complex coordinate plane.
So notice how this length here or this distance would be r, the radius.
And if we sketch a reference triangle, since x is negative 3 and y is positive 2,
we would label this leg negative 3 and this leg positive 2.
And angle theta would be this angle here.
Notice how theta terminates in the second quadrant.
So now let's go ahead and find r and theta.
So r equals the square root of x squared plus y squared, or in this case,
negative 3 squared plus 2 squared, which would be the square root of 13.
And now to find theta, tangent theta is equal to y divided
by x. We're using our reference triangle.
Notice how tangent theta would be equal to positive 2 divided by negative 3
or tangent theta equals negative two-thirds.
[ Writing on Board ]
So if we take the inverse tangent of both sides of the equation,
theta equals inverse tangent of negative two-thirds.
Now, let's [inaudible] the calculator and determine theta.
Let's use radians to determine the angle.
So we'll press "Mode."
Notice how we're already in radian mode, so we'll go back to the home screen.
And now we'll press "2nd Tangent" for our tangent or inverse tangent, negative two-thirds.
Close parenthesis and enter.
Now, remember the range for inverse tangent is from negative pi over 2
to positive pi over 2 radians here.
But we know our angle is in the second quadrant, so the calculator's giving us this angle here
in the fourth quadrant, so we'll have to add pi radians to find our angle.
Again, the calculator is giving us this negative angle here in the fourth quadrant
that has the same tangent function value, but we're looking for the angle
that terminates in the second quadrant.
So I'll have to add pi radians to the angle the calculator gave us.
So we'll take this value and add pi, which gives us r theta of approximately 2.5536.
Which means in polar form, our complex number, z,
is equal to r, which is the square root of 13.
Or if we want, 13 raised to the one half power times cosine
of 2.5536 radians plus i sine 2.5536 radians.
Now, remember this is z and we're looking for z to the one half.
So now we can apply Dumas' theorem where n is equal to one half.
So we'd have z to the one half equals the square root of 13 to the one half, which is equivalent
to 13 to the one fourth power times cosine of, again, if n is one half,
our angle is now one half times 2.5536 radians plus i times sine,
again, one half times 2.5536 radians.
And now we'll grab the calculator and get a decimal approximation for the real part
and the imaginary part of our square root where the real part would be this product
and the imaginary part would be this product.
So again, the square root of 13 raised to the one half is equivalent to 13 to the one-fourth,
but I'll go ahead and use this form.
So we'll have in parentheses the square root of 13, right arrow, close parenthesis,
raised to the power of one half or 1 divided by 2, right arrow, times cosine of one half
or 0.5 times theta, which is 2.5536 radians.
So the real part is approximately 0.55.
And now to find the imaginary part, we'll find the second product.
So again, we have the square root of 13 raised
to the one half power times -- we won't enter the i now.
We'll just enter sine, again, 0.5 times theta.
So the imaginary part is approximately 1.82i.
So plus 1.82i.
And now we have found the approximate value of the square root of negative 3 plus 21.
I hope you found this helpful. ------------------------------e7e39de5be9c--