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Question: If \(w\) is the complex cube root of unity, find the value of i. \(w + \dfrac{1}{w}\) ii. \({w^2...

If ww is the complex cube root of unity, find the value of
i. w+1ww + \dfrac{1}{w}
ii. w2+w3+w4{w^2} + {w^3} + {w^4}
iii. (1+w2)3{\left( {1 + {w^2}} \right)^3}
iv. (1−w−w2)3+(1−w+w2)3{\left( {1 - w - {w^2}} \right)^3} + {\left( {1 - w + {w^2}} \right)^3}

Explanation

Solution

Before solving the complex cube root questions, we need to know the basic properties of the complex cube root of the unity. Which tells that w3=1{w^3} = 1 which can also be written in the form of 1+w+w2=01 + w + {w^2} = 0 using these two equations, we can find the value of all the above options.

Complete step by step solution:
In the above question, we are given that ww is the complex cube root of unity which mean w3=1{w^3} = 1
or can be written as 1+w+w2=01 + w + {w^2} = 0
Or we can use it as 1+w=−w21 + w = - {w^2} or 1+w2=−w1 + {w^2} = - w
Now, finding the
i. w+1ww + \dfrac{1}{w} =w2+1w = \dfrac{{{w^2} + 1}}{w} (cross-multiplying)
=−ww=−1= \dfrac{{ - w}}{w} = - 1 (using the above written properties 1+w2=−w1 + {w^2} = - w)
ii. w2+w3+w4{w^2} + {w^3} + {w^4}
taking common and then using the above property 1+w+w2=01 + w + {w^2} = 0
=w2(1+w+w2)= {w^2}\left( {1 + w + {w^2}} \right)
=0= 0
iii. (1+w2)3{\left( {1 + {w^2}} \right)^3}
Using the property 1+w2=−w1 + {w^2} = - w
=(−w)3=−w3 =−1  = {\left( { - w} \right)^3} = - {w^3} \\\ = - 1 \\\ (using w3=1{w^3} = 1 )
iv. (1−w−w2)3+(1−w+w2)3{\left( {1 - w - {w^2}} \right)^3} + {\left( {1 - w + {w^2}} \right)^3}
Using w+w2=−1w + {w^2} = - 1 in first term and 1+w2=−w1 + {w^2} = - w in the second term
=(1+1)3+(−w−w)3 =(2)3[1−w3] =(2)3[1−1] =0  = {\left( {1 + 1} \right)^3} + {\left( { - w - w} \right)^3} \\\ = {\left( 2 \right)^3}\left[ {1 - {w^3}} \right] \\\ = {\left( 2 \right)^3}\left[ {1 - 1} \right] \\\ = 0 \\\
Note: Cube root of units means cube root of one. The roots are w,w2,w3(=1)w,{w^2},{w^3}( = 1) and the value of w is −1+32\dfrac{{ - 1 + \sqrt 3 }}{2}. We can also learn the properties of nth{n^{th}} root of unity, which gives us a general formula for any root of unity. This basically means that finding the root of 11n{1^{\dfrac{1}{n}}}. Most of the results are analogous to the cube root of unity and these results can be useful to solve complex problems.