what property or properties are these problems?

(#1)

The first problem could be an example of the distributive property / law.

a x (b + c) = (a x b) + (a x c)

Basically there are two ways to solve the problem:

(10-8) x (5) = (2) x (5) = 10

or

(10 - 8) x (5) = (10 x 5) - (8 x 5) = 50 - 40 = 10

The distributive law ensures that you will get the same answer either way.

(#2)

The second problem is most likely intended to be an example of the associative property / law:

(a+b)+c = a+(b+c) or in the case of multiplication (a x b) x c = a x ( b x c)

(Note that 5^2 means "five squared" or "to the second power")

3 x (5^2 x 1/12) = 3 x (25 x 1/12) = 3 x (25/12) = 25/4

or

3 x (5^2 x 1/12) = (3 x 5^2) x 1/12 = 75 x 1/12 = 75/12 = 25/4

you could also use the "commutative property" combined with the "associative property" to make it even easier: The "commutative property" is:

(a+b) = (b+a) or in the case of multiplication (a x b) = (b x a)

3 x (5^2 x 1/12) = 3 x (1/12 x 5^2) = (3 x 1/12) x 5^2 = 1/4 x 5^2 = 25/4

the commutative and associative laws look simple but are extremely powerful when combined. To illustrate what happened in the previous line.

let a=3, b=5^2 and c=1/12

1st) Use the commutative law

a x (b x c) = a x (c x b)

2nd) Use the associative law:

a x (c x b) = (a x c) x b

That way you can cancel out a and save some unnecessary multiplication.

(#3)

The final problem is the "Transitive law / property" which states:

If (a=b) and (b=c) then (a=c)

In this case the problem is saying since (4 times 2 equals 8) and (6 plus 2 also equals 8), we know that (4 times 2) must equal (6 plus 2).

These laws sound trivial, but we actually use it all the time without realizing it. If they weren't true then the mathematics we use in our day to day lives would not work.

(10-8)(5)=2(5) is true

3(5 to the second power x 1/12)- i have no earthly clue... what is that?

if 4x2=8 and 8=6+2 then YES 4x2=6+2

simple math except for the second one.. i have no idea