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If \( Z=\frac{(\sqrt{3}+i)^{3}(3 i+4)^{2}}{(8+6 i)^{2}} \), then \( |Z| \) is equal to
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Verified Answer
The correct answer is:
\( 12 \)
Given that, \( Z=\frac{(\sqrt{3}+i)^{3}(3 i+4)^{2}}{(8+6 i)^{2}} \)
So, \( |Z|=\frac{|\sqrt{3}+i|^{3}|3 i+4|^{2}}{|8+6 i|^{2}} \)
Now, \( |\sqrt{3}+i|=\sqrt{3+1}=2 \)
\( |3 i+4|=\sqrt{3^{2}+4^{2}}=5 \)
\( |8+6 i|=\sqrt{8^{2}+6^{2}}=10 \)
So, \( |Z|=\frac{|\sqrt{3}+i|^{3}|3 i+4|^{2}}{|8+6 i|^{2}} \)
Now, \( |\sqrt{3}+i|=\sqrt{3+1}=2 \)
\( |3 i+4|=\sqrt{3^{2}+4^{2}}=5 \)
\( |8+6 i|=\sqrt{8^{2}+6^{2}}=10 \)
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