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.