SATHEE: Complex Numbers 5 Question 19

Complex Numbers 5 Question 19

20.

If $x = a + b, y = a α + b β, z = a β + b α$ , where $α, β$ are complex cube roots of unity, then show that $x y z = a^{3} + b^{3}$ .

$(1979, 3 M)$

Show Answer

Solution:

Since, $α, β$ are the complex cube roots of unity.

$∴$ We take $α = ω$ and $β = ω^{2}$ .

Now, $x y z = (a + b) (a α + b β) (a β + b α)$

$= (a + b) [a^{2} α β + a b (α^{2} + β^{2}) + b^{2} α β]$

$= (a + b) [a^{2} (ω \cdot ω^{2}) + a b (ω^{2} + ω^{4}) + b^{2} (ω \cdot ω^{2})]$

$= (a + b) (a^{2} - a b + b^{2}) [∵ 1 + ω + ω^{2} = 0$ and $ω^{3} = 1]$

$= a^{3} + b^{3}$

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