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Question: If \(\omega \) is a complex cube root of unity, then \({\text{225 + (3}}\omega {\text{ + 8}}{\omega ...

If ω\omega is a complex cube root of unity, then 225 + (3ω + 8ω2)2 + (3ω2 + 8ω)2 = {\text{225 + (3}}\omega {\text{ + 8}}{\omega ^{\text{2}}}{{\text{)}}^{\text{2}}}{\text{ + (3}}{\omega ^{\text{2}}}{\text{ + 8}}\omega {{\text{)}}^{\text{2}}}{\text{ = }}

Explanation

Solution

Hint: The cube roots of unity can be defined as the numbers which when raised to the power of 3 gives the result as 1. In simple words, the cube root of unity is the cube root of 1 i.e.13\sqrt[3]{1}. According to the properties of the cube root of 1, the sum of its roots is zero. So, 1 + ω + ω2{\text{1 + }}\omega {\text{ + }}{\omega ^{\text{2}}}=0.

Complete step-by-step answer:

Given expression,
225 + (3ω + 8ω2)2 + (3ω2 + 8ω)2{\text{225 + (3}}\omega {\text{ + 8}}{\omega ^{\text{2}}}{{\text{)}}^{\text{2}}}{\text{ + (3}}{\omega ^{\text{2}}}{\text{ + 8}}\omega {{\text{)}}^{\text{2}}}
We know,
(a + b)2 = a2 + 2ab + b2{{\text{(a + b)}}^{\text{2}}}{\text{ = }}{{\text{a}}^{\text{2}}}{\text{ + 2ab + }}{{\text{b}}^{\text{2}}}
Also, 1, ω\omega , ω2{\omega ^{\text{2}}} are cube roots of unity.
According to the properties of the cube root of 1, the sum of its root is zero. So, 1 + ω + ω2{\text{1 + }}\omega {\text{ + }}{\omega ^{\text{2}}} =0
And ω3{\omega ^{\text{3}}}= 1.
Now, simplify the given expression
225 + (3ω)2 + (64ω4) + 48ω3 + 9ω4 + 64ω2 + 48ω3{\text{225 + (3}}\omega {{\text{)}}^{\text{2}}}{\text{ + (64}}{\omega ^{\text{4}}}{\text{) + 48}}{\omega ^{\text{3}}}{\text{ + 9}}{\omega ^{\text{4}}}{\text{ + 64}}{\omega ^{\text{2}}}{\text{ + 48}}{\omega ^{\text{3}}}
=225 + ω2(9 + 64) + ω3(48 + 48) + ω4(64 + 9){\text{225 + }}{\omega ^{\text{2}}}{\text{(9 + 64) + }}{\omega ^{\text{3}}}{\text{(48 + 48) + }}{\omega ^{\text{4}}}{\text{(64 + 9)}}
=225 + ω2(73) + ω3(96) + ω×ω3(73){\text{225 + }}{\omega ^{\text{2}}}{\text{(73) + }}{\omega ^{\text{3}}}{\text{(96) + }}\omega \times {\omega ^{\text{3}}}{\text{(73)}}
=225 + 73(ω + ω2) + 96{\text{225 + 73(}}\omega {\text{ + }}{\omega ^{\text{2}}}{\text{) + 96}}
=225+73(-1)+96
=248
Therefore, the value of the given expression is 248.

Note: The root of unity is a number which is complex in nature and gives 1 if raised to the power of a positive integer n. These roots are used in different branches and topics of maths like number theory. The cube roots of unity can be defined as the numbers which when raised to the power of 3 gives the result as 1. And there are a total of three cube roots of unity.