If ∣z_1 ∣ = 2, ∣z_2 ∣ = 3, ∣z_3 ∣ = 4 and ∣2z_1 + 3z_2 + 4z_3 ∣ = 4, then the expression ∣8z_2 z_3 + 27z_3 z_1 + 64z_1 z


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Question Number 20739 by Tinkutara last updated on 02/Sep/17

$If ∣ z_{1} ∣ = 2, ∣ z_{2} ∣ = 3, ∣ z_{3} ∣ = 4 and$ $∣ 2 z_{1} + 3 z_{2} + 4 z_{3} ∣ = 4, then the expression$ $∣ 8 z_{2} z_{3} + 27 z_{3} z_{1} + 64 z_{1} z_{2} ∣ equals$
	Answered by $@ty@m last updated on 02/Sep/17

	$∣ 8 z_{2} z_{3} + 27 z_{3} z_{1} + 64 z_{1} z_{2} ∣$ $=∣ 2 ∣ z_{1} ∣^{2} z_{2} z_{3} + 3 z_{1} ∣ z_{2} ∣^{2} z_{3} + 4 z_{1} z_{2} ∣ z_{3} ∣^{2} ∣$ $=∣ 2 z_{1} . {\bar{z}}_{1} z_{2} z_{3} + 3 z_{1} z_{2} . {\bar{z}}_{2} z_{3} + 4 z_{1} z_{2} z_{3} {\bar{z}}_{3} ∣ (∵∣ z ∣^{2} = z . \bar{z})$ $=∣ z_{1} z_{2} z_{3} ∣ . ∣ 2 {\bar{z}}_{1} + 3 {\bar{z}}_{2} + 4 {\bar{z}}_{3} ∣$ $=∣ z_{1} z_{2} z_{3} ∣ . ∣ 2 z_{1} + 3 z_{2} + 4 z_{3} ∣ (∵∣ z ∣=∣ \bar{z} ∣)$ $=∣ z_{1} ∣∣ z_{2} ∣∣ z_{3} ∣ \times 4$ $= 2 \times 3 \times 4 \times 4$ $= 96$
	Commented by Tinkutara last updated on 03/Sep/17

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