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2 <p>Last updated on<strong>August 5, 2025</strong></p>

3 <p>The cube root of Unity (or one) are the values which, when multiplied together, gives the original number Unity. The Cube Root of Unity is represented by ∛1, which actually have three roots→ 1,𝛚, 𝛚², which on multiplication together gives “1” as a product. 1×𝛚×𝛚²=1.</p>

4 <h2>What Is the Cube Root of Unity?</h2>

5 <p>What Is Algebra? 🧮 | Simple Explanation with 🎯 Cool Examples for Kids | ✨BrightCHAMPS Math</p>

6 <p>▶</p>

7 <p>As mentioned above, the<a>cube</a>root of Unity are 1,𝛚, 𝛚², where 1 is a real root, 𝛚 and 𝛚² are the imaginary roots. The essential features or properties of the cube root of Unity are:</p>

8 <p>The imaginary roots 𝛚 and 𝛚², when multiplied together, yields 1</p>

9 <p>𝛚×𝛚²= 𝛚³=1</p>

10 <p>The summation of the roots is zero → 1+𝛚+𝛚²=0.</p>

11 <p>The imaginary root 𝛚, when squared, is expressed as 𝛚², which is equal to another imaginary root.</p>

12 <p>Fact check: Do you know? The values of Cube root of (-1) are -1, -𝛚, and -𝛚² </p>

13 <h2>Finding the Cube Root of Unity</h2>

14 <p>Now, let us find the meaning of 𝛚 here. To find the<a>cube root</a>of Unity, we will make use of some algebraic<a>formulas</a>. We know that, the cube root of unity is represented as ∛1. Let us assume that ∛1= a so,</p>

15 <p>∛1= a</p>

16 <p>⇒ 1 = a3</p>

17 <p>⇒ a3- 1 = 0 </p>

18 <p> ⇒ (a - 1)(a2+a+1) = 0 [using a3-b3= (a - b)(a2+a.b+b2)]</p>

19 <p>⇒a - 1 =0</p>

20 <p>⇒ a= 1 …………..(1) </p>

21 <p>Again, a2+a+1 = 0</p>

22 <p> ⇒ a = (-1 ±√(12-4×1×1)) / 2×1</p>

23 <p>⇒ a = (-1 ±√(-3)) / 2</p>

24 <p>⇒ a = (-1 ± i√3) / 2</p>

25 <p>⇒ a = (-1 + i√3) / 2 …………(2)</p>

26 <p>Or</p>

27 <p>a = (-1 - i√3) / 2 …………(3)</p>

28 <p>From<a>equation</a>(1), (2), and (3), we get,</p>

29 <p>The roots are → 1, (-1 + i√3) / 2 and (-1 - i√3) / 2</p>

30 <p> Hence, 𝛚 = (-1 + i√3) / 2</p>

31 <p>𝛚2= (-1 - i√3) / 2 </p>

32 <h2>Common Mistakes and How to Avoid Them in the Cube Root of Unity</h2>

33 <p>some common mistakes with their solutions given:</p>

34 <h3>Explore Our Programs</h3>

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36 <h3>Problem 1</h3>

37 <p>Factorize m²+ mn + n²</p>

38 <p>Okay, lets begin</p>

39 <p>We know that, 1+𝛚+𝛚2=0</p>

40 <p>⇒ 𝛚+𝛚2= -1 ……….(1)</p>

41 <p>And, 𝛚3=1 …….(2)</p>

42 <p>So, m2+mn+n2</p>

43 <p>= m2 - (-1)mn +1× n2</p>

44 <p>= m2 - (𝛚+𝛚2)mn + 𝛚3× n2 [Using (1) and (2)]</p>

45 <p>= m2- mn𝛚- mn𝛚2+ n2𝛚3</p>

46 <p>= m(m-n𝛚) -n𝛚2(m-n𝛚)</p>

47 <p>= (m-n𝛚)(m-n𝛚2)</p>

48 <p>Answer : (m-n𝛚)(m-n𝛚2) </p>

49 <h3>Explanation</h3>

50 <p>We used the properties of the cube root of unity to factorise the expression. </p>

51 <p>Well explained 👍</p>

52 <h3>Problem 2</h3>

53 <p>Find 𝛚⁶⁶</p>

54 <p>Okay, lets begin</p>

55 <p>𝛚66</p>

56 <p>=(𝛚3)22</p>

57 <p>=(1)22</p>

58 <p>=1</p>

59 <p>Answer: 1 </p>

60 <h3>Explanation</h3>

61 <p>We used the property 𝛚3=1, and solved the expression. </p>

62 <p>Well explained 👍</p>

63 <h3>Problem 3</h3>

64 <p>Prove that (1+𝛚)³+(1+𝛚²)³ = -2</p>

65 <p>Okay, lets begin</p>

66 <p>We know that, 1+𝛚+𝛚2=0 ⇒1+𝛚= -𝛚2 ……….(1) And also, 1+𝛚2= -𝛚 ………(2)</p>

67 <p> LHS = (1+𝛚)3+(1+𝛚2)3</p>

68 <p>=(-𝛚2)3+(-𝛚)3 [Using (1) and (2)]</p>

69 <p>=(-𝛚6)+(-𝛚3) </p>

70 <p>= -(𝛚3)2 - (𝛚3)</p>

71 <p>=-(1)2 - 1 [using the property 𝛚3=1]</p>

72 <p>= -1-1</p>

73 <p>=-2</p>

74 <p>=RHS [proved] </p>

75 <h3>Explanation</h3>

76 <p>We proved the given expression to be true using properties of cube root if unity like 1+𝛚+𝛚2=0 and 𝛚3=1. </p>

77 <p>Well explained 👍</p>

78 <h3>Problem 4</h3>

79 <p>Prove that (1+𝛚-𝛚²)⁶= -64</p>

80 <p>Okay, lets begin</p>

81 <p>We know that, 1+𝛚+𝛚2=0</p>

82 <p> ⇒1+ 𝛚= -𝛚2 ……….(1)</p>

83 <p>And, 𝛚3=1 …….(2)</p>

84 <p>LHS</p>

85 <p>= (1+𝛚-𝛚2)6</p>

86 <p>=(-𝛚2-𝛚2)6 [using (1)]</p>

87 <p>=(-2𝛚2)6</p>

88 <p>=26 × (-𝛚2)6</p>

89 <p>=64× (-𝛚12)</p>

90 <p>= 64× (-(𝛚3)4)</p>

91 <p>= 64× (-(1)4)</p>

92 <p>= 64× (-1)</p>

93 <p>= -64</p>

94 <p>=RHS </p>

95 <h3>Explanation</h3>

96 <p>LHS=RHS</p>

97 <p>Hence proved</p>

98 <p>Well explained 👍</p>

99 <h2>FAQs for Cube Root of Unity</h2>

100 <h3>1.What is the cube root of unity rule?</h3>

101 <p> The cube root of unity has three roots → 1,𝛚, and 𝛚2, 1 is the real root, and 𝛚 and 𝛚2 are the imaginary roots. Also, the important properties of cube root of unity include: </p>

102 <p>1+𝛚+𝛚2=0</p>

103 <p>𝛚3=1</p>

104 <p>The<a>square</a>of the imaginary root omega, 𝛚 is the other imaginary root, 𝛚2 </p>

105 <h3>2.What is the expression of the cube root of unity ?</h3>

106 <p>The<a>expression</a>of cube root of unity is x3=1 or, x=∛1. We generally solve this expression by using the formula a3-b3= (a - b)(a2+a.b+b2). </p>

107 <h3>3.How do you find the cube root of unity?</h3>

108 <p>We can find the cube root of unity, i.e., the values of 𝛚 and 𝛚2 by solving the expression x3=1 or, x=∛1, using the formula a3-b3= (a - b)(a2+a.b+b2). We then land on to a value which deals with the Complex<a>numbers</a></p>

109 <h3>4. Is the cube root of unity 1?</h3>

110 <p>The cube root of unity is not only 1, it comprises two imaginary roots also, </p>

111 <p>𝛚 = (-1 + i√3) / 2 and 𝛚2= (-1 - i√3) / 2. </p>

112 <h2>Important Glossaries for Cube Root of Unity</h2>

113 <ul><li><strong>Omega -</strong>This is a symbol used for depicting the imaginary roots of the cube root of unity. It is represented by 𝛚. </li>

114 </ul><ul><li><strong>Complex Number -</strong>The numbers which are represented as m+i.n, where m and n are real numbers and “i”, known as iota, is an imaginary number. </li>

115 </ul><h2>Jaskaran Singh Saluja</h2>

116 <h3>About the Author</h3>

117 <p>Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.</p>

118 <h3>Fun Fact</h3>

119 <p>: He loves to play the quiz with kids through algebra to make kids love it.</p>