Why can't 5(6 * 5) be broken to 5(6) * 5(5) but 5 (6 + 8) can be broken to 5(6) + 5(8)

[CoolShank8]

So I was doing quadratics and I came across (5 + 6) ^ 2, but if I don't understand the basics I don't think I can progress,

 

[Dr.Peterson]

Your expression 5(6*5) means 5*(6*5), which is just 5*6*5; it's all multiplications, so the associative property applies, not the distributive property that applies to 5(6+5).

As for (5+6)^2, if all you need to do is to evaluate it, just add and then square: 5+6 = 11, and 11^2 = 121.

If there were a variable, you might want to expand (distribute), but there's no benefit in doing so here.

 

[Steven G]

In this case, (5+6)^2, I agree with Dr Peterson that expanding will not help. However that is not always true. Just consider 11^2 which you need to know to solve your problem. Suppose you did not know what 11^2 equals. You can realize 11 as 10 + 1. So 11^2 = (10 + 1)^2 = 10^2 + 2*10*1 + 1^2 = 100 + 20 + 1 = 121.

Similarly 19^2, for example, equals (20-1)^2 = 20^2 - 2*20*1 + 1^2 = 400 - 40 + 1 = 360 + 1 = 361

 

[Steven G]

Using the associate law we have 5*6*5 = 5*(6*5). Since in 5*6*5 there is no distributing the 5 and that is the same as 5*(6*5) then there is no distributing the 5 in 5*(6*5)

Here is another reason, you can do what is inside the parenthesis first, namely 6*5 which is 30 and then you have 5*(30). Now in 5*(30) there is not to distribute the 5 by. All you can do is multiply 5*30 and get 150. In on the other hand in 5*(6*5) you distribute the 5 you get (5*6)*(5*5) = 30*25 = 750. Since 750 is not 150 one of the methods must be wrong and it is not 5*(6*5) = 5*(30) = 150