Answer:
[tex]\sqrt{2} \,\sqrt{3} \,\sqrt{3} =3\,\sqrt{2}[/tex]
[tex]\sqrt{2} \,\sqrt{3} \,\sqrt{-3} =3\,i\,\sqrt{2}[/tex]
[tex]\sqrt{-2} \,\sqrt{-3} \,\sqrt{-3} =-3\,i\,\sqrt{2}[/tex]
4) [tex]\sqrt{2} \,\sqrt{-3} \,\sqrt{-3} =-3\,\sqrt{2}[/tex]
Step-by-step explanation:
Let's work on each expression at a time, recalling the definition of the imaginary unit [tex]\sqrt{-1} =i[/tex], and the properties of radical multiplication: [tex]\sqrt{a} \,\,\sqrt{b} =\sqrt{a\,b}[/tex]
1) [tex]\sqrt{2} \,\sqrt{3} \,\sqrt{3} =\sqrt{2} \,\sqrt{3\,*\,3}=\sqrt{2} \,\sqrt{3^2}=\sqrt{2} \,*\,3=3\,\sqrt{2}[/tex]
2) [tex]\sqrt{2} \,\sqrt{3} \,\sqrt{-3} =\sqrt{2} \,\sqrt{3}\,\,\sqrt{3} \,\,\sqrt{-1} =\sqrt{2} \,\sqrt{3^2}\,\,i=3\,i\,\sqrt{2}[/tex]
3) [tex]\sqrt{-2} \,\sqrt{-3} \,\sqrt{-3} =\sqrt{-1} \,\,\sqrt{2}\,\, \sqrt{-1} \,\,\sqrt{3}\,\,\sqrt{-1} \,\,\sqrt{3}=i^3\,\sqrt{2} \,\,\sqrt{3^2} =(-1)\,i\,\sqrt{2}\,\,3=-3\,i\,\sqrt{2}[/tex]
4) [tex]\sqrt{2} \,\sqrt{-3} \,\sqrt{-3} =\sqrt{2} \,\,\sqrt{-1} \,\,\sqrt{3} \,\,\sqrt{-1} \,\,\sqrt{3} \,=\sqrt{2} \,\,i^2 \,\,\sqrt{3} \,\,\sqrt{3} \,\,=(-1)\,\sqrt{2} \,\,\sqrt{3^2} =-3\,\sqrt{2}[/tex]
