Find the fourth roots of 5(1 + i√3)
Real and imaginary numbers combine to form complex numbers. The imaginary component, I (iota), indicates a square root of -1. The imaginary portion of a complex number is i. a + ib is a typical representation of complex numbers in their rectangular or standard form. For example, 100 + 25i is a complex number in which 100 represents the real part and 25i represents the imaginary part.
Polar Form Representation of Complex Numbers
To represent a complex number, the polar coordinates of the real and imaginary components are written here. represents the angle at which the number line is inclined to the real axis, i.e. the x-axis. The length indicated by the line is known as its modulus, and it is represented by the letter r in the alphabet. The real and imaginary components are represented by a and b, respectively, while the modulus is represented by OP = r in the diagram below.
The Pythagoras theorem is to be used to get the length r. Trigonometric ratios can be used to calculate arguments. The polar form of a complex number of the type z = a + ib is represented as follows:
r = Modulus[cos(argument) + isin(argument)]
Alternatively, z = r[cosθ + isinθ]
In this case, r = and θ = tan-1{b/a}.
Calculating Roots of Complex Numbers
DeMoivre’s Theorem can be used to simplify higher-order complex numbers. It can be used to determine the roots of complex numbers as well as expand complex numbers according to their exponent.
Given: , then its roots are:
Where, k lies between 0 and n – 1 and n is the exponent or radical.
Find the fourth roots of 5(1 + i√3). Leave in trigonometrirm.
Solution:
Modulus of the given number =
= 10
Argument = tan-1[5√3/ 5] = π/3.
Thus, the polar form of 5 + 5√3i =
According to DeMoivre’s formula, all the nth roots of a complex number are given by:
, where k lies between 0 and n – 1 and n is the exponent or radical.
Here, r = 10, θ = π/3 and n = 4.
Find the 4 roots by substituting the values of k as 0, 1, 2 and 3 respectively.
- For k = 0, z =
=
- For k = 1, z =
=
- For k = 2, z =
=
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- For k = 3, z =
=
Thus, the four roots of 5(1 + i√3) are
and
.
Similar Problems
Question 1: Find the cube roots of -2 – 2√3i.
Solution:
r =
= √(16) = 4, θ = 4π/ 3.
According to DeMoivre’s formula, all the nth roots of a complex number are given by:
, where k lies between 0 and n – 1 and n is the exponent or radical.
Find the 3 roots by substituting the values of m as 0, 1 and 2 respectively,
- For m = 0, z =
= 0.27 + 1.56i
- For m = 1, z =
= −1.49 − 0.54i
- For m = 2, z =
= 1.21 −1.02i
Thus, the roots are 0.27 + 1.56i, −1.49 − 0.54i and 1.21 – 1.02i.
Question 2: Find the fifth roots of 32 + 0i.
Solution:
Modulus =
= 32.
Argument = θ = tan-1(0/ 32) = 0.
According to DeMoivre’s formula, all the nth roots of a complex number are given by:
, where k lies between 0 and n – 1 and n is the exponent or radical.
Find the 5 roots by substituting the values of m as 0, 1, 2, 3 and 4.
- For m = 0, z =
= 2
- For m = 1, z =
= 0.62 + 1.9i
- For m = 2, z =
= −1.62 + 1.18i
- For m = 3, z =
= −1.62 − 1.18i
- For m = 4, z =
= 0.62 − 1.9i
Thus, the roots are 2, 0.62 + 1.9i, -1.62 + 1.18i, -1.62 – 1.18i and 0.62 – 1.9i. &
;
Question 3: Find the fourth roots of -8√3 + 8i.
Solution:
Polar form =
We have k = 2, n = 4 and θ = 5π/ 6.
According to DeMoivre’s formula, all the nth roots of a complex number are given by:
, where k lies between 0 and n – 1 and n is the exponent or radical.
Find the 4 roots by substituting the values of m as 0, 1, 2 and 3 respectively.
- For m = 0, z =
= 1.58 + 1.21i.
- For m = 1, z =
= −1.21 + 1.58i.
- For m = 2, z =
= −1.58 −1.21i.
- For m = 3, z =
= 1.21 − 1.58i.
Thus, the four roots of z are 1.58 + 1.21i, −1.21 + 1.58i, −1.58 −1.21i and 1.21 − 1.58i.
Question 4: Find the sixth root of -27i. Leave in trigonometric form.<
ong>Solution:
Modulus =
Argument = θ = tan-1(-27/ 0) = π/2.
Polar form =
According to DeMoivre’s formula, all the nth roots of a complex number are given by:
, where k lies between 0 and n – 1 and n is the exponent or radical.
Find the 6 roots by substituting the values of m as 0, 1, 2, 3, 4 and 5.
- For m = 0, z =
=
- For m = 1, z =
=
- For m = 2, z =
=
- For m = 3, z =
=
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- For m = 4, z =
=
- For m = 5, z =
=
Question 5: Find the fourth root of 81i. Leave in trigonometric form.
Solution:
Polar form =
We have k = 81, n = 4 and θ = π/ 2.
According to DeMoivre’s formula, all the nth roots of a complex number are given by:
, where k lies between 0 and n – 1 and n is the exponent or radical.
Find the 4 roots by substituting the values of m as 0, 1, 2 and 3 respectively.
- For m = 0, z =
=
- For m = 1, z =
=
- For m = 2, z =
=
- For m = 3, z =