Answer:
[tex]\sqrt{-63}=3i\sqrt{7}[/tex]
[tex]\sqrt{-156}=2i\sqrt{39}[/tex]
[tex]6\sqrt{-12}=12i\sqrt{3}[/tex]
[tex]-5\sqrt{-75}=-25i\sqrt{3}[/tex]
Step-by-step explanation:
An imaginary number is a non-zero real number multiplied by the imaginary unit [tex]i[/tex].
[tex]\boxed{\begin{minipage}{4 cm}\begin{center}\underline{Imaginary Number Rule}\end{center}\\\begin{center}$\sqrt{-1}=i$\end{center}\end{minipage}}[/tex]
[tex]\boxed{\begin{minipage}{4 cm}\begin{center}\underline{Radical Rule}\end{center}\\\begin{center}$\sqrt{a \cdot b}=\sqrt{a} \sqrt{b}$\end{center}\end{minipage}}[/tex]
[tex]\begin{aligned}\sqrt{-63} & = \sqrt{9 \cdot -1 \cdot 7}\\& = \sqrt{9}\sqrt{-1}\sqrt{7}\\& = 3i\sqrt{7}\end{aligned}[/tex]
[tex]\begin{aligned}\sqrt{-156} & = \sqrt{4 \cdot -1 \cdot -39}\\& = \sqrt{4} \sqrt{-1} \sqrt{39}\\& = 2i\sqrt{39}\end{aligned}[/tex]
[tex]\begin{aligned}6 \sqrt{-12} & = 6 \sqrt{4 \cdot -1 \cdot -3} \\& = 6 \sqrt{4} \sqrt{-1} \sqrt{3} \\& = 6 \cdot 2 i \sqrt{3} \\& = 12i \sqrt{3}\end{aligned}[/tex]
[tex]\begin{aligned}-5 \sqrt{-75} & = -5 \sqrt{25 \cdot -1 \cdot -3} \\& = -5 \sqrt{25} \sqrt{ -1 } \sqrt{3} \\& = -5 \cdot 5i \sqrt{3} \\& = -25i \sqrt{3}\end{aligned}[/tex]
Learn more about imaginary numbers here:
https://brainly.com/question/26344541
