z^4 + 81 = 0 (solve)

change it to polar, then take the root.

z^4= 81@180
z= 3@180/4 + n90 where n=0, 1, 2, 3

check:
z= 3@180/4 + 3*90=3@315
z^4=81@(4*315)=81@1260= 91@180=-81
you can check it at the other roots also.

this is what i did:

z^4 = - 81
z (81 cis 180)^1/4
81 ^ 1/4 cis 1/4 * 180
3 cis 45 therefore 360/4 = 90*
4 solutions:
3 cis 45*
3 cis 135*
3 cis - 135*
3 cis - 45*
???

That is it. You get four roots, the same as mine. I did not use positive and negative angles, just positive. That is a preference I have. It is easier for me to think of rotation in one direct, so your -45 is 315 for me.

thanks for the help, a bit confusing trying to read explanations for maths over the internet but hey we got the right answers. :)

It does take a while to understand the ASCII labels for math. Fractions are the most difficult..

Glad you got it.

To solve the equation z^4 + 81 = 0, we can follow these steps:

Step 1: Rewrite the equation in polar form by converting z^4 = 81@180.
Step 2: Find the fourth root of both sides of the equation.
Step 3: Determine the values of z by using the formula z = (81^1/4)@((180 + 360n)/4), where n is an integer ranging from 0 to 3.

After solving, we obtain four solutions for z:

1. z = 3@45 degrees (or 3 cis 45 degrees)
2. z = 3@135 degrees (or 3 cis 135 degrees)
3. z = 3@225 degrees (or 3 cis -135 degrees)
4. z = 3@315 degrees (or 3 cis -45 degrees)

I apologize if the explanation was a bit confusing. Understanding mathematical expressions in written form can indeed be challenging. However, I'm glad we arrived at the correct answers. If you have any more questions or need further assistance, feel free to ask!

To solve the equation z^4 + 81 = 0, you can start by changing it to polar form. This is done by expressing z as r cis θ, where r is the magnitude and θ is the angle.

First, rewrite the equation as z^4 = -81. Then, represent -81 in polar form as 81 cis 180, where 81 is the magnitude and 180 is the angle.

To find the fourth root of z^4, you can take the fourth root of the magnitude (81^(1/4)) and divide the angle by 4.

In this case, 81^(1/4) equals 3. Now, divide 180 by 4 to get 45, which gives the angle of the first root.

Since you're looking for four solutions, you can add n * 90 to the angle, where n is 0, 1, 2, or 3.

So, the four solutions for z are:

1. z = 3 cis (45 + 0 * 90)
2. z = 3 cis (45 + 1 * 90)
3. z = 3 cis (45 + 2 * 90)
4. z = 3 cis (45 + 3 * 90)

Simplifying each of these solutions:
1. z = 3 cis 45
2. z = 3 cis 135
3. z = 3 cis -135
4. z = 3 cis -45

These are the four roots of the equation z^4 + 81 = 0 in polar form.

If you want to check if these solutions are correct, you can substitute each value of z back into the original equation and verify that it equals zero.