Answer:
1) [tex]\sqrt{-48}=4\sqrt{3} \ i[/tex]
2) [tex]\sqrt{-63}=3\sqrt{7} \ i[/tex]
3) [tex]\sqrt{-72}=6\sqrt{2} \ i[/tex]
4) [tex]\sqrt{-24}=4\sqrt{6} \ i[/tex]
5) [tex]\sqrt{-84}=2\sqrt{21} \ i[/tex]
6) [tex]\sqrt{-99}=3\sqrt{11} \ i[/tex]
Step-by-step explanation:
We need to simply the following :
Note: For all questions given we know that [tex]\sqrt{-1}=i[/tex] using it in all the given questions.
1) [tex]\sqrt{-48}[/tex]
Factors of 48 are: 2x2x2x2x3
[tex]\sqrt{-48}\\=\sqrt{48}\sqrt{-1} \\=\sqrt{2\times2\times2\times2\times3}\ i\\= \sqrt{2^2\times2^2\times3}\ i\\=2\times2\sqrt{3} \ i \\=4\sqrt{3} \ i[/tex]
2) [tex]\sqrt{-63}[/tex]
Factors of 63 are: 3x3x7
[tex]\sqrt{-63}\\=\sqrt{63}\sqrt{-1} \\=\sqrt{3\times3\times7}\ i\\= \sqrt{3^2\times7}\ i\\=3\sqrt{7} \ i[/tex]
3) [tex]\sqrt{-72}[/tex]
Factors of 72 are: 2x2x2x3x3
[tex]\sqrt{-72}\\=\sqrt{72}\sqrt{-1} \\=\sqrt{2\times2\times2\times3\times3}\ i\\= \sqrt{2^2\times3^2\times2}\ i\\=2\times3\sqrt{2} \ i \\=6\sqrt{2} \ i[/tex]
4) [tex]\sqrt{-24}[/tex]
Factors of 24 are: 2x2x2x3
[tex]\sqrt{-24}\\=\sqrt{24}\sqrt{-1} \\=\sqrt{2\times2\times2\times3}\ i\\= \sqrt{2^2\times2\times3}\ i\\=2\sqrt{2\times3} \ i \\=2\sqrt{6} \ i[/tex]
5) [tex]\sqrt{-84}[/tex]
Factors of 84 are: 2x2x3x7
[tex]\sqrt{-84}\\=\sqrt{84}\sqrt{-1} \\=\sqrt{2\times2\times3\times7}\ i\\= \sqrt{2^2\times3\times7}\ i\\=2\sqrt{21} \ i[/tex]
6) [tex]\sqrt{-99}\\[/tex]
Prime factors of 99 are: 3x3x11
[tex]\sqrt{-99}\\=\sqrt{99}\sqrt{-1} \\=\sqrt{3\times3\times11}\ i\\= \sqrt{3^2\times11}\ i\\=3\sqrt{11} \ i \\=3\sqrt{11} \ i[/tex]
