Monomials

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

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)

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).

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 ${2}+{1}+{3}={6}$.

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.