# Monomials

Monomial is an algebraic expression, that can have the following 3 "parts":

• Number (it is called coefficient of monomial)
• Variables, raised to non-negative integer powers
• Operations of multiplication (they "separate" variables)

$\color{green}{\underbrace{15}_{\text{number (coefficient)}}}\color{blue}{\overbrace{\cdot}_{\text{multiplication}}} \color{red}{\underbrace{x^2}_{\text{variable}}} \color{blue}{\overbrace{\cdot}_{\text{multiplication}}} \color{red}{\underbrace{y^3}_{\text{variable}}}$

Any of the above 3 "parts" can be skipped.

Examples of monomials:

• ${15}$ (just number is a monomial, variables and multiplication are skipped)
• ${{x}}^{{2}}$ (it can be thought, that there is no coefficient, but it is there! It is 1.)
• ${15}{x}$ (valid monomial with one variable and multiplication sign, that is not written)
• ${2}{{x}}^{{2}}{{y}}^{{3}}$ (monomial with two variables)

Note, that addition, subtraction and division are not allowed for "separating" variables, only for writing coefficient.

More examples:

• $\frac{{x}}{{2}}=\frac{{1}}{{2}}{x}$ (coefficient is $\frac{{1}}{{2}}$)
• ${\left({2}+\sqrt{{{2}}}\right)}{x}{{y}}^{{2}}$ (coefficient involves roots)

Now, let's see examples of expressions, that are not monomials:

• ${2}{x}+{y}$ (addition is used to "separate" variables)
• $\frac{{y}}{{{x}}^{{2}}}$ (division of variables is not allowed)
• ${{2}}^{{x}}$ (variable exponent is not allowed)
• ${2}{{m}}^{{\frac{{1}}{{3}}}}{{n}}^{{-{2}}}$ (negative and fractional exponents are not allowed).

Degree of the monomial is the sum of exponents of all variables it contains.

Since constant monomial doesn't contain variables, its degree equals 0.

Example. Degree of ${35}$ is ${0}$.

Example. Degree of ${2}{x}$ is ${1}$.

Example. Degree of $-{5}{{y}}^{{2}}{x}{{z}}^{{3}}$ is ${2}+{1}+{3}={6}$.

Monomials are called like terms if they have the same variables to the same power.

For example, ${2}{\color{red}{{{{x}}^{{5}}{{y}}^{{7}}}}}$ and $-{4}{\color{red}{{{{x}}^{{5}}{{y}}^{{7}}}}}$ are like terms, but ${2}{{x}}^{{3}}$ and ${2}{x}{{y}}^{{2}}$ are not.

Exercise 1. Determine, whether the following is a monomial: ${2}{{x}}^{{2}}{y}$?

Exercise 2. Determine, whether the following is a monomial: $\frac{{2}}{\sqrt{{{3}}}}{x}{y}$?

Exercise 3. Determine, whether the following is a monomial: ${2}\frac{{x}}{{y}}$?
Exercise 4. Find degree of the monomial ${14}{{x}}^{{3}}$.
Exercise 5. Find degree of the monomial $-{9}{{x}}^{{3}}{{a}}^{{11}}{{p}}^{{7}}$.