**Solve Equations in Complex Numbers With Cube Roots of Unity :**

Here we are going to see some example problems of solving equations in complex numbers with cube roots of unity.

**Question 1 :**

Solve the equation z^{3} + 27 = 0.

**Solution :**

z^{3} + 27 = 0

z^{3} = -27

z^{3} = (-1 ⋅ 3)^{3}

z = [(-1 ⋅ (3)^{3}] ^{1/3}

= 3 (-1)^{1/3}

Polar form of -1 :

-1^{ } = 3[cos π + i sin π]

^{ } = [cos(2kπ + π) + i sin (2kπ + π)]

^{ } = [cos π(2k + 1)) + i sin π(2k + 1)]

(-1)^{1/3}^{ } = [cos π(2k + 1)) + i sin π(2k + 1)]^{1/3}

(-1)^{1/3 } = [cos (π/3)(2k + 1)) + i sin (π/3)(2k + 1)]

k = 0, 1, 2

If k = 0

= [cos (π/3)(2k + 1)) + i sin (π/3)(2k + 1)]

= 3 cis (π/3)

If k = 1

= 3 [cos π + i sin π]

= -3

If k = 2

= [cos (5π/3) + i sin (5π/3)]

= 3 cis (5π/3)

Let us look into the next problem on "Solve Equations in Complex Numbers With Cube Roots of Unity"

**Question 2 :**

If ω ≠ 1 is a cube root of unity, show that the roots of the equation (z −1)^{3} + 8 = 0 are −1, 1− 2ω, 1− 2ω^{2}

**Solution :**

(z −1)^{3} + 8 = 0

(z −1)^{3} = -8

(z −1) = (-8)^{1/3 }

(z −1) = -2 ⋅ (1) ^{1/3 }

z = -2 ⋅ (1) ^{1/3 }+ 1

z = 1 - 2 ⋅ (1) ^{1/3 }

Cube root of 1 are 1, ω, ω^{2}

z = 1 - 2 ⋅ 1

z = 1 - 2 = -1

z = 1 - 2 ⋅ ω

z = 1 - 2ω

z = 1 - 2 ⋅ ω^{2}

z = 1 - 2ω^{2}

Let us look into the next problem on "Solve Equations in Complex Numbers With Cube Roots of Unity"

**Question 3 :**

Find the value of

**Solution :**

If k = 1,

= cos 2π/9 + i sin 2π/9 ----(1)

If k = 2,

= cos 4π/9 + i sin 4π/9 ----(2)

If k = 3,

= cos 6π/9 + i sin 6π/9 ----(3)

If k = 4,

= cos 8π/9 + i sin 8π/9 ----(4)

...................

By adding all these, we get

= cis (π/9) (2 + 4 + 6 + 8 + 10 + 12 + 14 + 16)

= cis (π/9) 2(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8)

= cis (72π/9)

= cis 8π

= cos 8π + i sin 8π

= 1 + i(0)

= 1

**Question 4 :**

If ω ≠ 1 is a cube root of unity, show that

(i) (1 − ω + ω^{2})^{6} + (1 + ω − ω^{2})^{6} = 128.

(ii) (1 − ω)(1 + ω^{2})(1 + ω^{4})(1 + ω^{8}).............(1 + ω^{2^11}) = 1

**Solution :**

**(i) (1 − ω + ω ^{2})^{6} + (1 + ω − ω^{2})^{6} = 128.**

**L.H.S: **

** = (1 + ω ^{2 }**

** ****= (- ω ^{ }**

** ****= (- 2ω) ^{ }**

** = 64 ****ω**^{6} ** + 64****ω ^{12}**

** = 64 (****ω**^{3})^{2 }+ **64 (****ω**^{3})^{4}

** = 64 + 64**

** = 128 **

**R.H.S**

**Hence proved.**

(ii) (1 − ω)(1 + ω^{2})(1 + ω^{4})(1 + ω^{8}).............(1 + ω^{2^11}) = 1

L.H.S

(1 − ω)(1 + ω^{2})(1 + ω^{4})(1 + ω^{8})(1 + ω^{16}) (1 + ω^{32}) (1 + ω^{64})

(1 + ω^{128})(1 + ω^{256})(1 + ω^{512})(1 + ω^{1024})(1 + ω^{2048})

First 2 terms are = (1 − ω)(1 + ω^{2})

3rd and 4th terms :

(1 + ω^{4})(1 + ω^{8}) = (1 + ω)(1 + ω^{2})

5^{th} and 6^{th} terms :

(1 + ω^{16})(1 + ω^{32}) = (1 + ω)(1 + ω^{2})

Similarly by grouping these terms, we get

= [(1 + ω)(1 + ω^{2})]^{6}

= [1 + ω^{2 }+ ω + ω^{3}^{ }]^{6}

= [0 + ω^{3}^{ }]^{6}

= 1

Hence proved.

**Question 5 :**

If z = 2 - 2i, find rotation of z by θ radians in the counter clock wise direction about the origin when

(i) θ = π/3 (ii) θ = 2π/3 (iii) θ = 3π/2

**Solution :**

**z = 2 - 2i **

α = tan^{-1}|(-2)/2|

α = tan^{-1}|(-2)/2|

After having gone through the stuff given above, we hope that the students would have understood, "Solve Equations in Complex Numbers With Cube Roots of Unity".

Apart from the stuff given in this section "Solve Equations in Complex Numbers With Cube Roots of Unity", if you need any other stuff in math, please use our google custom search here.

HTML Comment Box is loading comments...

**WORD PROBLEMS**

**HCF and LCM word problems**

**Word problems on simple equations **

**Word problems on linear equations **

**Word problems on quadratic equations**

**Area and perimeter word problems**

**Word problems on direct variation and inverse variation **

**Word problems on comparing rates**

**Converting customary units word problems **

**Converting metric units word problems**

**Word problems on simple interest**

**Word problems on compound interest**

**Word problems on types of angles **

**Complementary and supplementary angles word problems**

**Markup and markdown word problems **

**Word problems on mixed fractrions**

**One step equation word problems**

**Linear inequalities word problems**

**Ratio and proportion word problems**

**Word problems on sets and venn diagrams**

**Pythagorean theorem word problems**

**Percent of a number word problems**

**Word problems on constant speed**

**Word problems on average speed **

**Word problems on sum of the angles of a triangle is 180 degree**

**OTHER TOPICS **

**Time, speed and distance shortcuts**

**Ratio and proportion shortcuts**

**Domain and range of rational functions**

**Domain and range of rational functions with holes**

**Graphing rational functions with holes**

**Converting repeating decimals in to fractions**

**Decimal representation of rational numbers**

**Finding square root using long division**

**L.C.M method to solve time and work problems**

**Translating the word problems in to algebraic expressions**

**Remainder when 2 power 256 is divided by 17**

**Remainder when 17 power 23 is divided by 16**

**Sum of all three digit numbers divisible by 6**

**Sum of all three digit numbers divisible by 7**

**Sum of all three digit numbers divisible by 8**

**Sum of all three digit numbers formed using 1, 3, 4**

**Sum of all three four digit numbers formed with non zero digits**