simplify and state restrictions on the variable
a)(4+squareroot2)(5-squareroot8)
b)2squareroot5+squareroot5
c)3squareroot7-5squareroot7
d)squareroot35/squareoot7
1) I don't know how to make it pretty, so I hope this works: First foil it:
$\displaystyle (4+sqrt(2))(5-sqrt(8))$
$\displaystyle (4)(5) - 4sqrt(8) + 5sqrt(2) - sqrt(2)sqrt(8)$
Now you need to simplify $\displaystyle sqrt(8)$ and $\displaystyle sqrt(8)sqrt(2)$
This is how I do it: Take sqrt(8) and break it into it's factors, so you have:
$\displaystyle sqrt(8)=sqrt(2*4)=sqrt(2)*sqrt(4)=2*sqrt(2)$
Make sense? So if you do that we have:
$\displaystyle 20 - 4*2*sqrt(2) + 5sqrt(2) - sqrt(16)$
Now you can just simplify:
$\displaystyle 16 - 3sqrt(2)$
2) This one is a cinch:
$\displaystyle 2sqrt(5) + sqrt(5) = 3sqrt(5)$
Do you see why?
3) This is the same as 2.
4) You need to break sqrt(35) into it's factors:
$\displaystyle sqrt(35) = sqrt(5*7) = sqrt(5)sqrt(7)$
Now we have:
$\displaystyle sqrt(35)/sqrt(7) = sqrt(5)sqrt(7)/sqrt(7) = sqrt(5)$
The two sqrt(7)s just cancel out.