Answer:
Part 1) [tex]2\sqrt{6}[/tex]
Part 2) [tex]4\sqrt{5}[/tex]
Part 3) [tex]36\sqrt{10}[/tex]
Part 4) [tex]15\sqrt{5}[/tex]
Part 5) [tex]20\sqrt{5}[/tex]
Part 6) [tex]9\sqrt{2}[/tex]
Step-by-step explanation:
Part 1) we have
[tex]\sqrt{24}[/tex]
we know that
[tex]24=(2^{3})(3)[/tex]
Remember that
[tex]\sqrt{a^{2}}=a[/tex]
substitute
[tex]\sqrt{(2^{3})(3)}=\sqrt{(2^{2})(2)(3)}=2\sqrt{6}[/tex]
Part 2) we have
[tex]\sqrt{80}[/tex]
we know that
[tex]80=(2^{4})(5)[/tex]
Remember that
[tex]\sqrt{a^{2}}=a[/tex]
substitute
[tex]\sqrt{(2^{4})(5)}[/tex]
we have that
[tex](2^{4})=(2^{2})^{2}=4^{2}[/tex]
substitute
[tex]\sqrt{(4^{2}(5)}=4\sqrt{5}[/tex]
Part 3) we have
[tex]12\sqrt{90}[/tex]
we know that
[tex]90=(2)(3^{2})(5)[/tex]
Remember that
[tex]\sqrt{a^{2}}=a[/tex]
substitute
[tex]12\sqrt{(2)(3^{2})(5)}=12\sqrt{(3^{2})(5)(2)}=(12)(3)\sqrt{(5)(2)}=36\sqrt{10}[/tex]
Part 4) we have
[tex]3\sqrt{125}[/tex]
we know that
[tex]125=(5^{3})[/tex]
Remember that
[tex]\sqrt{a^{2}}=a[/tex]
substitute
[tex]3\sqrt{(5^{3})}=3\sqrt{(5^{2})(5)}=(3)(5)\sqrt{5}=15\sqrt{5}[/tex]
Part 5) we have
[tex]2\sqrt{500}[/tex]
we know that
[tex]500=(2^{2})(5^{3})[/tex]
Remember that
[tex]\sqrt{a^{2}}=a[/tex]
substitute
[tex]2\sqrt{(2^{2})(5^{3})}=2\sqrt{(2^{2})(5^{2})(5)}=(2)(2)(5)\sqrt{5}=20\sqrt{5}[/tex]
Part 6) we have
[tex]\sqrt{162}[/tex]
we know that
[tex]162=(2)(3^{4})[/tex]
Remember that
[tex]\sqrt{a^{2}}=a[/tex]
substitute
[tex]\sqrt{(2)(3^{4})}=\sqrt{(2)(3^{2})^{2}}=(3^{2})\sqrt{(2)}=9\sqrt{2}[/tex]
