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-2-equals-

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

Thank you very much Sir!