Please help and explain?

Question

Perform the indicated operations:
√[50] + 2√[32] – √[8]


Multiply:
(5√[2] + 3)(√[2] – 2√[3])

Rationalize the denominator:
2
__________
√[6] – √[5]

Answer 1

√[50] + 2√[32] – √[8]
=√(25)(2) + 2√(16)(2) - √(4)(2)
= (√25)(√2) + 2(√16)(√2) - (√4)(√2)
= 5√2 + 8√2 - 2√2
= 11√2


(5√[2] + 3)(√[2] – 2√[3])
= (5√2)(√2) + (3)(√2) - (5√2)(2√3) - (3)(2√3)
= (5)(2) + 3(√2) - (5*2)(√2*3) - 6√3
= 10 + 3√2 - 10√6 - 6√3


2 / (√6 - √5)
= (2 / (√6 - √5)) ((√6 + √5) / (√6 + √5))
= (2(√6 + √5)) / ((√6 - √5)(√6 + √5))
= (2(√6 + √5)) / (6 - 5)
= 2(√6 + √5)

Answer 2

1. If you have any rational number under a square root, if you factor the number under the square root you have not changed it's value.

So... rt[50] is the same as rt[25]*rt[2] is the same as 5*rt[2]
2*rt[32] = 2*rt[16]*rt[2]= 2*4*rt[2] = 8*rt[2]
rt[8] = rt[4]*rt[2] = 2*rt[2]

Now you have 5*rt[2] + 8*rt[2] - 2*rt[2].
Since you have a rt[2] in each you can perform the operation and come up with 11rt[2].

2. Mulitply (5*rt[2] + 3) (rt[2]-2*rt[3])

Here you foil, so (5*rt[2])(rt[2]) + (3)(rt[2]) - (2*rt[3])(5*rt[2]) - (3)(2*rt[3]) --- remember when multiplying two square roots you multiply the numbers beneath the square root signs.
So when you perform the multiplication you get:
5*rt[4] + 3*rt[2] - 10*rt[6] -6*rt[3].

Then simplify and combine any like terms: 10 + 3*rt[2] - 10*rt[6] - 6*rt[3].

3. Rationalize the denominator in (2)/(rt[6] - rt[5]).
In order to rationalize the denominator you need to perform an action that removes the square root signs.

Recall the difference of squares which tells us if we multiply (x-y) by (x+y) we get x^2-y^2. So if we multiply the denominator by (rt[6] + rt[5]) we get (6-5) or 1.

Additionally, whatever action we take in the denominator we also have to take in the numerator so we have to multiply 2 by (rt[6] + rt[5]).

So the numerator becomes 2*rt[6] + 2*rt[5].

With the denominator now 1, the answer is 2*rt[6] + 2*rt[5].

Answer 3

5√2 + 8√2 -2√2 = 11√2

10 - 10√6 + 3√2 - 6√3 (is there a √3 missing)

2(√6 + √5)

Answer 4

1) sqrt(25*2) +2Sqrt(16*2) - sqrt(4*2)
= 5sqrt(2) + 8sqrt(2) - 2sqrt(2)
= sqrt(2)[5+8-2]
= sqrt(2)[11]
= 11sqrt(2)

2) [5sqrt(2) + 3][sqrt(2) - 2sqrt(3)]
= 5*2 -10sqrt(6) + 3sqrt(2) -6sqrt(3)
= 10 - 10sqrt(6) + 3sqrt(2) -6sqrt(3)

3) 2 / Sqrt(6) -Sqrt(5)
Multiplying and dividing by[Sqrt(6) +Sqrt(5)]
=2[Sqrt(6) +Sqrt(5)] / [Sqrt(6) -Sqrt(5)] [Sqrt(6) +Sqrt(5)]
=2[Sqrt(6) +Sqrt(5)] / [Sqrt(6)^2 -Sqrt(5)^2]
= 2[Sqrt(6) +Sqrt(5)] / 6-5
=2[Sqrt(6) +Sqrt(5)] / 1
= 2[Sqrt(6) +Sqrt(5)]