Adding Monomials

A monomial, or term, is a number symbol made of a numeral, a letter, a product of letters, or a numeral times one or more letters. For example, 5, -a, bx, -4cy^3, \tfrac{3}{5}a^2x^3y, 2^7, and 2^{x+y} are all terms. When it won't cause confusion, we also call things like (a+b), 3(x-y), 5\sqrt{x}, and \sqrt{a-x} terms, since each could be replaced by a single letter.

The literal part of a term is the part made of letters. Similar terms (also called like terms) are plain numbers, the same root of a number, or terms with exactly the same literal part. So a, 4a, and -10a are similar, and so are a^2x, -3a^2x, and 7a^2x. Dissimilar terms have different literal parts: 7a^2x, 3ax, and 5ax^2 are all different.

Adding similar terms

We know that 6 acres and 3 acres make 9 acres. In the same way 6a + 3a = 9a, 6a + (-3a) = 3a, and 5xy + 6xy = 11xy. A term like -y or xy has an unwritten coefficient of 1: -y means -1y, and xy means +1xy.

Rule. To add similar terms, find the algebraic sum of the coefficients and write it in front of the common literal part.

Adding dissimilar terms

Order doesn't matter in addition: 2 + 3 = 3 + 2, and a + b + c = b + c + a = c + a + b. Dissimilar terms like 6d and 7c can't be merged, so their sum is just written 6d + 7c. Adding 6d and -7c gives 6d + (-7c), which is the same as 6d - 7c.

Rule. To add dissimilar terms, write them one after another in any order, each with its proper sign.

When a sum has both kinds, use both rules: merge the similar terms and list the rest.

Typing tip: type \sqrt{a} as sqrt(a), and you can type a whole parenthesis like 9(a+b).