If x = a + b, y = aγ + bβ and z = aβ +...

MathematicsCube Roots and nth Roots of UnityJEE Advanced 1978Easy

If $x = a + b, y = aγ + b β$ and $z = a β + bγ$ where $γ$ and $β$ are the complex cube roots of unity, show that $x y z = a^{3} + b^{3}$ .

Visualized Solution (Hindi)

Identifying the Roots .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } γ and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } β

Let the complex cube roots of unity be $ω$ and $ω^{2}$ .
Given: $γ = ω$ and $β = ω^{2}$ .
Key Properties:
1. $ω^{3} = 1$
2. $1 + ω + ω^{2} = 0 ⟹ ω + ω^{2} = - 1$

Setting up the Product .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } x y z

Substitute $x, y, z$ in the product expression:
$x y z = (a + b) (aω + b ω^{2}) (a ω^{2} + bω)$

Expanding the Complex Terms

Multiply the last two brackets:
$(aω + b ω^{2}) (a ω^{2} + bω) = a^{2} ω^{3} + ab ω^{2} + ab ω^{4} + b^{2} ω^{3}$

Simplifying Powers of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ω

Use $ω^{3} = 1$ and $ω^{4} = ω$ :
$= a^{2} (1) + ab ω^{2} + abω + b^{2} (1)$
$= a^{2} + ab (ω^{2} + ω) + b^{2}$

Applying the Sum Property

Substitute $ω^{2} + ω = - 1$ :
$= a^{2} + ab (- 1) + b^{2}$
$= a^{2} - ab + b^{2}$

The Final Identity .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a^{3} + b^{3}

Combine with the first factor:
$x y z = (a + b) (a^{2} - ab + b^{2})$
Using the identity $a^{3} + b^{3} = (a + b) (a^{2} - ab + b^{2})$ :
$x y z = a^{3} + b^{3}$
Hence Proved.

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