**8ax + 3a + 40cx + 18c -- What rule did I use to switch the two middle terms? **(grouping like terms rule)

**a(8x + 3) + 40cx + 18c -- What rule did I use to separate the 'a' from the two terms in the parentheses? **a/a

a(8x + 3) + c(40x + 18) -- Again, what rule? c/c

Your

**purpose** was to group terms

**with a common factor,** but the

**name **of the rule that you used is the commutative law of addition, meaning

\(\displaystyle j + k = k + j.\)

The rule you used in the other two steps was the distributive rule of multiplication over addition, meaning

\(\displaystyle km + kn = k(m + n).\)

I am in the minority by believing that memorizing the names for these rules is far less important than knowing the rules and how to use them.

Here are some of the most basic rules.

\(\displaystyle a + b = b + a.\) Additive commutativity.

\(\displaystyle a * b = b * a.\) Multiplicative commutativity.

\(\displaystyle a + (-\ a) = 0.\) Additive inverses.

\(\displaystyle a + 0 = a \implies a + (b - b) = a.\) Additive identity.

\(\displaystyle a * \dfrac{1}{a} = 1 \implies \dfrac{a}{a} = 1.\) Multiplicative inverses.

\(\displaystyle a * 1 = a \implies a * \dfrac{b}{b} = a.\) Multiplicative identity.

\(\displaystyle a * 0 = 0.\)

\(\displaystyle ab + ac = a(b + c).\) Distributivity.

\(\displaystyle ab = 0 \implies a = 0 \text { or } b = 0 \text { or } a = 0 = b.\) Zero product property.

There are other rules of numbers, but a lot of algebra is just using the rules above to put expressions into a more useful form.