more square roots!

[Guest]

okay so right in the middle of my geometry book is a lesson on square roots. and the only thing i learned about square roots was like the square root of 25 is 5, and if it wasn't a whole number i left it unsquared. lol so how do you do things like

the square root of 200. or 5 times the sqaure root of 18. and another fun one, 4 over the square root of 2. (do you know what i mean). what about the square root of 72 plus the square root of 75 minus the square root of 48. more fun ones... solve for x if
(the square root of 5) squared plus (the square root of 11) squared= x squared.
okay and one more... x squared = (5 times the square root of three) squared plus (the square root of 5) squared????????? i'm sorry if you don't get what i'm saying but i don't have a square root key on my keyboard. lol, if you can help Thank You soooooooo much.

 

[pka]

Study these until you see how it works.
\(\displaystyle \L
\sqrt {75} = \sqrt {25 \cdot 3} = \sqrt {25} \cdot \sqrt 3 = 5\sqrt 3\)

\(\displaystyle \L
5\sqrt {18} = 5\sqrt {3^2 \cdot 2} = 15\sqrt 2\)

\(\displaystyle \L
\sqrt {200} = \sqrt {2^3 \cdot 5^2 } = 10\sqrt 2\)

 

[Guest]

okay, i think i kinda got i but how do i do the one with fractions, like 4 over the square root of 2?

 

[pka]

\(\displaystyle \L
\frac{4}{{\sqrt 2 }} = \left( {\frac{4}{{\sqrt 2 }}} \right)\left( {\frac{{\sqrt 2 }}{{\sqrt 2 }}} \right) = \frac{{4\sqrt 2 }}{2} = 2\sqrt 2\)