Monomials and Operations on them

Monomial is caled such expression, that contains the numbers, natural powers of variables and their products and doesn′t contain any other operations on numbers and variables.

For example, `3a*(2.5a^3)`, `(5ab^2)*(0.4c^3d)`, `x^2y*(-2z)*0.85` - monomials, whereas the expressions `a+b, (ab)/c` aren′t monomials.

Any monomial can be transformed to the standard form, i.e. to represent as product of numerical multiplier, that stay at the first place and powers of the different variables. The numerical multiplier of monomial, that written in the standard form, is called coefficient of monomial. The sum of exponents of all variables is called the power of monomial. If between two monomials to write multiplication sign, then we will obtain monomial, that is called the product of given monomials. When we raise the monomial to the natural power we also obtain monomial. The result we usually transform to the standard form.

The transformation of monomial to the standard form, the multiplication of monomials is identical transformation.

Example 1. We should transform the monomial `3a*(2.5a^3)` to the standard form:

`3a*(2.5a^3)=(3*2.5)*(a*a^3)=7.5a^4`.

Example 2. We should multiply monomials `24ab^2cd^3` and `0.3a^2b^3c`.

`24ab^2cd^3*(0.3a^2b^3c)=(24*0.3)*(a*a^2)*(b^2*b^3)*(c*c)*d^3=7.2a^3b^5c^2d^3` .

Example 3. We should raise the monomial `(-3ab^2c^3)` to the fourth power.

`(-3ab^2c^3)^4=(-3)^4*a^4*(b^2)^4*(c^3)^4=81a^4b^8c^12` .

The monomials, that are transformed to the standard form are called similar, if they are differ only by coefficients or dont′t differ. The similar monomials we can add and subtract, and as result we obrain monomial again, that is similar to original (sometimes we obtain 0). Addition and subtraction of similar monomials are called transformation of similar terms.

Example 4. We should add `5x^2yz^3` and `-8x^2yz^3` .

`5x^2yz^3+(-8x^2yz^3)=(5+(-8))x^2yz^3=-3x^2yz^3` .