plz help find the fourth root of -16,giving the result in form of a+jb
and also 5th root of -1,giving the result in polar form plz help thanks
?-16 = ?(16 cis ?) = ?16 cis ?/4 = 2(1/?2 + 1/?2 i) = ?2 + ?2 i
That is the 1st root. There is an equivalent root in each quadrant. See
http://www.wolframalpha.com/input/?i=x%5E4+%3D+-16
Work the 5th root the same way, using de Moivre's theorem.
sir how did u get the argument? And the modulus?
Putoyanah
here is another way, to answer your argument and modulus question.
-1=1@(180+n*360
fifth root of that is 1@36+n(72)
so the roots are
1@36
1@108
1@180
and you can do the last two. Now that angle is on the complex plane, If you want it in real,imag format, it is converted by
real=1*cosTheta+i*1*sinTheta
if you convert all those, you will see a pattern of symmetry, clockwise and counterclockwise.
To find the fourth root of -16 and express it in the form a+jb (rectangular form), we can use the following steps:
Step 1: Express -16 in rectangular form:
-16 = -16 + 0i
Step 2: Find the magnitude of -16:
|m| = √((-16)^2 + 0^2) = √(256) = 16
Step 3: Find the principal argument (θ) of -16:
θ = atan(0/(-16)) = 0 degrees
Step 4: Calculate the fourth root of -16 in the polar form:
-16^(1/4) = (16∠(0/4 + (360/4)*k))^(1/4), where k is an integer
Step 5: Simplify the angles:
16∠(0 + (90*k))^(1/4) = 2∠(0 + (90*k/4)) = 2∠(22.5*k), where k is an integer
Step 6: Calculate the fourth root of -16 in the rectangular form:
a + jb = 2(cos(22.5*k) + j*sin(22.5*k)), where k is an integer
Now, let's find the 5th root of -1 and express it in polar form:
Step 1: Express -1 in rectangular form:
-1 = -1 + 0i
Step 2: Find the magnitude of -1:
|m| = √((-1)^2 + 0^2) = √(1) = 1
Step 3: Find the principal argument (θ) of -1:
θ = atan(0/(-1)) = 0 degrees
Step 4: Calculate the fifth root of -1 in the polar form:
(-1)^(1/5) = (1∠(0/5 + (360/5)*k))^(1/5), where k is an integer
Step 5: Simplify the angles:
1∠(0 + (72*k))^(1/5) = 1∠(0 + (72*k/5)) = 1∠(14.4*k), where k is an integer
Step 6: Express the result in polar form:
The fifth root of -1 in polar form is 1∠14.4 degrees.