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|$z_1$|=2,|$z_2$|=3,|$z_3$|=4 and |$2z_1+3z_2+4z_3$|=9 then the absolute value of $8z_2z_3+27z_3z_1+64z_1z_2$ must be equal to?

($z_1,z_2,z_3$ are complex numbers)

I tried manipulating with the conjugates and stuff...but not being able to figure out the right technique to solve..help please!!

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HINT: Try to multiply the expression $|2\bar{z_1}+3\bar{z_2}+4\bar{z_3}|$ by the product $|z_1z_2z_3|$. What is the value of $|2\bar{z_1}+3\bar{z_2}+4\bar{z_3}|$?

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  • $\begingroup$ I guess you meant divide and not multiply. $\endgroup$ – user220382 Mar 28 '15 at 5:51
  • $\begingroup$ No, I meant multiply. I fixed it, I wanted to consider the absolute value of the right hand side product. $\endgroup$ – Daniel Mar 28 '15 at 5:53
  • $\begingroup$ I got your one...its 216..the answer.. $\endgroup$ – user220382 Mar 28 '15 at 5:53
  • $\begingroup$ Arey thats called taking common..yes btw your hint was really good one..thanks a lot :-) $\endgroup$ – user220382 Mar 28 '15 at 5:53
  • $\begingroup$ You're welcome @SanchayanDutta. $\endgroup$ – Daniel Mar 28 '15 at 6:00
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Given expression equals $|z_1z_2z_3||8/z_1+27/z_2+64/z_3|$ =$|z1||z2||z3||2z_1+3z_2+4z_3|$=2.3.4.9=216

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