Operations with monomials

Example

$4x^{4}y$ and ${\displaystyle \frac{1}{5}}{x}^{4}y$

Other examples of similar monomials are:

$5x^{2}yh$ and ${\displaystyle \frac{6}{7}}{x}^{2}yh$

$x^2$ and $3x^2$

We have to remark that the order in which the variables appear is not relevant.

Example

$3x^{5}y+2x^{5}y=(3+2)x^{5}y=5x^{5}y$

$4x^3+6x^3=(4+6)x^3=10x^3$

$3xyh+11xyh=(3+11)xyh=14xyh$

Example

$3x^{5}y-2x^{5}y=(3-2)x^{5}y=1x^{5}y=x^{5}y$

$4x^3-6x^3=(4-6)x^3=-2x^3$

$3xyh-11xyh=(3-11)xyh=-8xyh$

Example

If we multiply the monomials $3x^{2}y$, ${\displaystyle \frac{3}{4}}zy$

The product of the coefficients is $$3\cdot {\displaystyle \frac{3}{4}}={\displaystyle \frac{9}{4}}$$

And that of the independent terms are $$(x^{2}y)\cdot (zy)=x^{2}yzy=x^2{y}^{2}z$$

So, the final result is $${\displaystyle \frac{9}{4}}{x}^2{y}^{2}z$$

Example

In the same way, if we multiply ${\displaystyle \frac{3}{4}}{x}^{6}z$, ${\displaystyle \frac{16}{7}}{z}^{2}y$

The product of the coefficients is $${\displaystyle \frac{3}{4}}\cdot {\displaystyle \frac{16}{7}}={\displaystyle \frac{12}{7}}$$

And that of the independent terms are $$(x^{6}z)\cdot (z^{2}y)=x^{6}z{z}^{2}y=x^6{z}^{3}y$$

So, the final result is $${\displaystyle \frac{12}{7}}{x}^6{z}^{3}y$$

Example

${\displaystyle \frac{3x^{2}y}{2xy}}={\displaystyle \frac{3}{2}}\cdot {\displaystyle \frac{x^{2}y}{xy}}={\displaystyle \frac{3}{2}}x$

${\displaystyle \frac{3x^{2}y}{2x^{4}y}}={\displaystyle \frac{3}{2}}\cdot {\displaystyle \frac{x^{2}y}{x^{4}y}}={\displaystyle \frac{3}{2}}{\displaystyle \frac{1}{x^2}}$

${\displaystyle \frac{3x^2}{2xz}}={\displaystyle \frac{3}{2}}\cdot {\displaystyle \frac{x^2}{xz}}={\displaystyle \frac{3}{2}}{\displaystyle \frac{x}{z}}$

Example

${\displaystyle \frac{x^{5}z}{5x^{3}z}}={\displaystyle \frac{1}{5}}\cdot {\displaystyle \frac{x^{5}z}{x^{3}z}}={\displaystyle \frac{1}{5}}{x}^2$

${\displaystyle \frac{7x^{3}z}{5x^{3}z}}={\displaystyle \frac{7}{5}}\cdot {\displaystyle \frac{x^{3}z}{x^{3}z}}={\displaystyle \frac{7}{5}}\cdot 1={\displaystyle \frac{7}{5}}$

${\displaystyle \frac{2h^7}{7h^3}}={\displaystyle \frac{2}{7}}\cdot {\displaystyle \frac{h^7}{h^3}}={\displaystyle \frac{2}{7}}{h}^4$

Example

Identical variables in the numerator and the denominator, some of them with a superior degree in the denominator.

${\displaystyle \frac{xz}{2x^{3}z}}={\displaystyle \frac{1}{2}}\cdot {\displaystyle \frac{xz}{x^{3}z}}={\displaystyle \frac{1}{2}}{\displaystyle \frac{1}{x^2}}={\displaystyle \frac{1}{2x^2}}$

${\displaystyle \frac{3x^{3}y}{x^3{y}^4}}={\displaystyle \frac{3}{1}}\cdot {\displaystyle \frac{x^{3}y}{x^3{y}^4}}=3\cdot {\displaystyle \frac{1}{y^3}}={\displaystyle \frac{3}{y^3}}$

${\displaystyle \frac{4x}{x^3}}={\displaystyle \frac{4}{1}}\cdot {\displaystyle \frac{x}{x^3}}=4\cdot {\displaystyle \frac{1}{x^2}}={\displaystyle \frac{4}{x^2}}$

Example

Cases with different variable in the denominator.

${\displaystyle \frac{4x^4{y}^2}{3h}}={\displaystyle \frac{4}{3}}\cdot {\displaystyle \frac{x^4{y}^2}{h}}$

${\displaystyle \frac{2x^5}{3x^6}}={\displaystyle \frac{2}{3}}\cdot {\displaystyle \frac{x^5}{x^6}}={\displaystyle \frac{2}{3}}\cdot {\displaystyle \frac{1}{x}}={\displaystyle \frac{2}{3x}}$

${\displaystyle \frac{1}{t}}$

It must be said that in the last example the degree of the monomial of the numerator is not relevant at all: it does not matter how big it is since if a new variable appears in the denominator, the result will always be a rational fraction.

Example

$(2x{)}^2=2^2\cdot x^2=4x^2$

$(3x{y}^2{)}^3=3^3\cdot (x{y}^2{)}^3=27\cdot x^3\cdot (y^2{)}^3=27x^3{y}^6$

$({\displaystyle \frac{1}{2}}xy{z}^3{)}^4=({\displaystyle \frac{1}{2}}{)}^4\cdot (xy{z}^3{)}^4={\displaystyle \frac{1}{2^4}}\cdot x^4\cdot y^4\cdot (z^3{)}^4={\displaystyle \frac{1}{16}}\cdot x^4\cdot y^4\cdot z^{12}$

In the first example of all, we must be aware that:

$(-2x{)}^2=(-2{)}^2\cdot x^2=4x^2$

But

$(-2x{)}^3=(-2{)}^3\cdot x^3=-8x^3$

So, if a monomial with negative coefficient is raised to an even exponent, the result will be positive; if it is raised to an odd exponent, it will be negative.

Let's see some more examples:

$(-{\displaystyle \frac{1}{2}}x{y}^2{)}^5=(-{\displaystyle \frac{1}{2}}{)}^5\cdot (x{y}^2{)}^5=-{\displaystyle \frac{1}{2^5}}\cdot x^5\cdot y^{10}=-{\displaystyle \frac{1}{32}}\cdot x^5\cdot y^{10}$

$(-{\displaystyle \frac{1}{2}}x{y}^2{)}^4=(-{\displaystyle \frac{1}{2}}{)}^4\cdot (x{y}^2{)}^4={\displaystyle \frac{1}{2^4}}\cdot x^4\cdot y^8={\displaystyle \frac{1}{16}}\cdot x^4\cdot y^8$

$(-{\displaystyle \frac{1}{3}}{x}^{3}h{)}^3=(-{\displaystyle \frac{1}{3}}{)}^3\cdot (x^{3}h{)}^3=-{\displaystyle \frac{1}{3^3}}\cdot x^9\cdot h^3=-{\displaystyle \frac{1}{27}}\cdot x^9\cdot h^3$

$(-{\displaystyle \frac{1}{3}}{x}^{3}h{)}^2=(-{\displaystyle \frac{1}{3}}{)}^2\cdot (x^{3}h{)}^2={\displaystyle \frac{1}{3^2}}\cdot x^6\cdot h^2={\displaystyle \frac{1}{9}}\cdot x^6\cdot h^2$