The complex number from the table are
[tex]1+\sqrt{-3}, \ 4-3 \sqrt{-16}, \ \frac{3+2 \sqrt{-9}}{7}, \ 9 \sqrt{-\frac{7}{5}}[/tex]
Solution:
Let us solve and identify the complex number.
(A) [tex]5-\sqrt{\frac{9}{4}}[/tex]
[tex]5-\sqrt{\frac{9}{4}}= 5-\sqrt{\frac{3^2}{2^2}}[/tex]
[tex]= 5-\frac{3}{2}[/tex]
= 3.5
This is not a complex number.
(B) [tex]1+\sqrt{-3}[/tex]
[tex]1+\sqrt{-3}=1+\sqrt{-1\times 3}[/tex]
We know that [tex]\sqrt{-1} =i[/tex].
[tex]1+\sqrt{-3}=1+\sqrt{3}i[/tex]
This is a complex number.
(C) [tex]4-3 \sqrt{-16}[/tex]
[tex]4-3 \sqrt{-16}=4-3 \sqrt{-1\times 4^2}[/tex]
[tex]4-3 \sqrt{-16}=4-3\times4 \sqrt{-1}[/tex]
We know that [tex]\sqrt{-1} =i[/tex].
[tex]4-3 \sqrt{-16}=4-12i[/tex]
This is a complex number.
(D) [tex]\frac{2-\sqrt{12}}{5}[/tex]
[tex]\frac{2-\sqrt{12}}{5}=\frac{2-2\sqrt{3}}{5}[/tex]
There is no –1 in the root.
This is not a complex number.
(E) [tex]\frac{3+2 \sqrt{-9}}{7}[/tex]
[tex]\frac{3+2 \sqrt{-9}}{7}=\frac{3+2 \sqrt{-1\times 3^2}}{7}[/tex]
We know that [tex]\sqrt{-1} =i[/tex].
[tex]=\frac{3+6i}{7}[/tex]
This is a complex number.
(F) [tex]9 \sqrt{-\frac{7}{5}}[/tex]
[tex]9 \sqrt{-\frac{7}{5}}=9 \sqrt{-1 \times \frac{7}{5}}[/tex]
We know that [tex]\sqrt{-1} =i[/tex].
[tex]9 \sqrt{-\frac{7}{5}}=9 \sqrt{ \frac{7}{5}}i[/tex]
This is a complex number.
Hence the complex number from the table are
[tex]1+\sqrt{-3}, \ 4-3 \sqrt{-16}, \ \frac{3+2 \sqrt{-9}}{7}, \ 9 \sqrt{-\frac{7}{5}}[/tex]
