for all real numbers a and b, a(b*c) = ab *ac
This statement is true for all real numbers and is known as the Commutative Property. It not only applies to multiplication, but to addition as well: a + b = b + a
The other case you mentioned a(b*c) = ab*ac is false.
There are certain other properties that are central to understanding algebra and working with real numbers that you may also learn depending on your level:
Assiociative property: Given real numbers a, b, and c: (a +b) + c = a + (b +c) also, (a*b)*c = a*(b*c)
Distributive property: Given real numbers a, b, and c: a*(b + c) = a*b + a*c
This property can be shown by applying the definition/concept of multiplication:
a*(b + c) means that we have "a" groups of (b + c) hence,
a*(b + c) = (b + c) + ... + (b + c) <- a times, we are adding an "a" number of (b + c)'s.
Next we ask the questions how many b's do we have and how many c's do we have? Well applying basic counting we know that there an "a" number of b's, and an "a" number of c's. in other words,
a*(b + c) = (b + c) + ... + (b + c) [a times] = a*b + a*c
Identity: Pretty states that there is a real number for addition and there is a real number for multiplication that doesn't interact with any other number for the respective operation. For addition the number is 0. 0 is know as the additive identity. For multiplication the number is 1, 1 is known as the multiplicative identity.
Additive inverse: The number that you add to a given real number to get the additive identity 0. For example -1 is the additive inverse of 1, and 5/2 is the additive inverse of -5/2. 0 is the additive inverse of 0.
Multiplicative inverse: The number that you multiply to a given real number to get the multiplicative identity. In almost every case it is the reciprical of that number. For example:
1/2 is the multiplicative inverse of 2, -3 is the multiplicative inverse of -1/3, as before 1 is the multiplicative inverse of 1. HOWEVER, 0 is the only real number with out a multiplicative inverse. There is no real number you can multiply by 0 to get 1. This is why we can't divide by 0, and this is why we say once we understand left and right cancelation that when zero is in the denominator the fraction is "undefined".
Left Cancelation: Given that an operation of addition or multiplication is taking place you can "undo" by using its inverse.
a + 4 = 10
a + 4 + -4 = 10 + -4
a + 0 = 6
a = 6
b * 7 = 21
b * 7 * (1/7) = 21 * (1/7)
b * 1 = 3
b = 3
The same can pretty much be said without lost of generality for right cancelation when we are subtracting and dividing on both sides we are really adding and dividing by the additive and multiplicative inverse.
Trichotemy Law: given two arbitrary real numbers a and b, one out of the following three relationships must take place: either a < b, a = b, a > b.
If you are not majoring in mathematics, the first five is pretty much all you need to know.