How can you use the properties of real numbers to simplify algebraic expressions

A bit of a broad question, but I'll walk through an example of an unsimplified algebraic expression and walk through the properties used to simplify it. There are a lot of properties you apply in your head without even thinking about it! We start with this expression:



The first property we'll apply is the distributive property, which states in its simplest case that, given three real numbers a, b, and c:

a(b + c) = ab + ac

If x is a real number, then we know that (x + 4) must be a real number too, since the sum of any two real numbers is itself a real number. We can treat this (x + 4) term the same way we treat that number a, and distribute it to the x and the 7, obtaining:



In the next step, we'll reuse the distributive property to further expand our expression, but we have to take note of a subtle detail first:

We originally defined the distributive property with the expression a(b+c); years of experience with the properties of multiplication might have conditioned you to view that expression as equivalent to (b+c)a, but that fact rests upon the application of another property: the commutative property of multiplication, which states in its simplest case that, for any two numbers a and b, ab = ba. a(b+c) and (b+c)a might not be totally equivalent statements, but they have equivalent values. Returning to the problem, we can use the distributive property and the commutative property to expand our expression:



From here, we can use the associative property of addition to regroup our terms, allowing us to combine the 4x and the 7x in our next step. In case you forgot, the associative property, in its simplest form, states that, for any three numbers a, b, and c:

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

This is what allows us to write expressions like a + b + c without parentheses; the associative property tells us that the order we add the numbers up isn't important.

Regrouping the terms in our expression, we get:



To combine the 4x and 7x terms, we again use the distributive property. Note that, by the commutative property of multiplication, the equation

a(b + c) = ab + ac

is equivalent to

(b+c)a = ba + ca

Here, our a is x, and our b and c are 4 and 7. Which means that:



Finally, we have our simplified expression:



Seeing the properties applied step-by-step in this way really gives you an appreciation for how foundational they are to algebra!