Lesson 2: Still More Arithmetic

Property

Like terms are any terms that are exactly alike in their variable factors.
The numerical factor in a term is called the numerical coefficient, or just the coefficient of the term.

Examples

• In the expression $8a + 3a - b$, the terms $8a$ and $3a$ are like terms because they both have the variable factor $a$. The term $-b$ is not like them.
• The terms $5xy$ and $-2xy$ are like terms. The terms $5x$ and $-2xy$ are not like terms because their variable factors, $x$ and $xy$, are different.
• In the term $15z$, the number $15$ is the numerical coefficient. In a term like $x$, the coefficient is understood to be $1$. For a term like $-y$, the coefficient is $-1$.

Explanation

Like terms are terms that have the exact same variable part, including any exponents. You can think of them as being the same 'type' of item, which allows you to group them together through addition or subtraction.

Property

We can combine like powers of the same variable. When we add like terms, we do not alter the exponent; only the coefficient of the power changes. For example:

Different powers of the same variable are not like terms and cannot be combined. For example, $8x^2 - 3x$ cannot be simplified.

Examples

• The terms $7a^3$ and $4a^3$ are like terms, so they can be combined: $7a^3 - 4a^3 = 3a^3$.

• The expression $5w^2 + 3w^3$ cannot be simplified because $w^2$ and $w^3$ are not like terms.

Property

First Law of Exponents

Second Law of Exponents