could you walk me thorugh these problems

Question

simplfy
square root 18 - 3 times sqaure root of 50 plus 5 times the square root of 80 - 2 times the square root of 125

I'm having troubles with the square root of 80 all of the other radicals can be simplfified with a 2 in the radical except for 80 so can you walk me through that problem

also how do i do this
(square root 5 - square root 8)^2

once i know how to do that problem i can probably figure out how to do this one which I don't know how to do either

(square root 71 - square root 21)(square root 71 + square root 21)

also how do I simplfy something that looks likethis

1 + (1/square root 2) all over quanitity(1 - (1/square root2)

Thanks for all the help once I know how to do these problems I shouldn't need anymore help thanks for taking the time to help me please walk me through how to simplify these types of problems

Answer 1

On the first: simplify all to haveing a factor sqrt5

on the second, use the FOIL method:
(sqrt5-sqrt8)^2=5-sqrt40-sqrt40+8 combine terms.
on the next, use foil (If you don't recognize it as the difference of two squares.
On the last, multipy the numberation and the denominator by (1+1/sqr2)

Then, it becomes two foil problems, just like the other two.

Answer 2

Your questions would be easier to answer if you'd write them in more conventional math notation. Here is how to do two of them.

(sqrt71 - sqrt21)(sqrt 71 + sqrt 21)
= 71 - (sqrt21)(sqrt 71) + (sqrt21)(sqrt 71) - 21 = 50

Just remember the general rule that
(a+b)(a-b) = a^2 -b^2, and you could have written that down right away.

For the next one, let's try to get rid of the fracxtions. First multiply numerator and denominator by sqrt2. That results in
[1 + (1/sqrt2)]/[1 - (1/sqrt2]
= (sqrt2 +1)/(sqrt2 -1)
Now multiply numerator and denominator by sqrt2 + 1, and that becomes
(sqrt2 +1)^2/(2-1) = 2 + 2sqrt2 + 1
= 3 + 2 sqrt2