## Summary

IDefinition, domain and range, graphical representationIIProperties and basic operationsIIIEquations with monomials## Definition, domain and range, graphical representation

### Monomial

A monomial function is a function of the following form:

**f(x)=ax^n**

a is a real number and n is a whole number.

The following functions are monomials:

- f(x)=2x^2
- g(x)=x
- h(x)=-17x^{101}

### Degree of a monomial

Let *a* be a real number and *n* be a whole number. In the monomial ax^n, the exponent n is referred to as the degree of the monomial.

Let *f* be the monomial such that f\left(x\right)=3x^5. The degree of *f* is 5.

Every constant function is by definition a monomial function. If b is real number, then:

b=bx^0

The domain of a monomial function is all real numbers.

Consider the following function:

f(x)=2x^3

The domain of f(x) is all real numbers.

The range of a monomial function depends on whether or not the exponent is even or odd. Let f(x)=ax^n be a monomial function and assume that a>0. Then the range of f(x) is:

- All real numbers if n is odd.
- All nonnegative numbers if n is even.

Consider the following function:

f(x)=2x^3

The range of f(x) is all real numbers.

Consider the following function:

f(x)=2x^2

The range of f(x) is [0,\infty).

Let f(x)=ax^n be a monomial function, the graph of f(x) will resemble the following:

## Properties and basic operations

If a and b are real numbers, then:

**ax^n+bx^n=(a+b)x^n **

**ax^n-bx^n=(a-b)x^n **

Therefore the sum of and difference of two monomials of the same degree is a monomial.

3x^2+17x^2=19x^2

3x^2-17x^2=-14x^2

The sum of two monomials of different degrees is not a monomial.

For example, the sum of the monomials x^2+7x^3 does not simplify to a monomial.

If a and b are real numbers, then:

**ax^n\cdot bx^m=abx^{n+m}**

Therefore the product of any two monomials is a monomial.

The product of the monomials 7x^3 and 3x^5 is:

7x^3\cdot 3x^5=21x^{3+5}=21x^{8}

When multiplying two monomials, the exponent of the new monomial is found by adding the previous exponents, not by multiplying the exponents.

## Equations with monomials

Let *a* be a real number and then consider the following equation:

x^n=a

If n is an odd number, then the equation has a unique solution denoted:

** x=a^{\frac{1}{n}} **

Let's consider the equation x^3=27.

3 is odd so the equation has a unique solution.

-3 is the unique number such that (-3)^{3}=-27.

We denote:

(-27)^{1/3}=-3

Let *a* be a real number and then consider the following equation:

x^n=a

If n is an even number then the equation has a two solutions:

**One positive solution, which is denoted by a^{\frac{1}{n}}.****The second solution: -a^{\frac{1}{n}}.**

Let's consider the equation x^2=16.

2 is even so the equation has two solutions.

4 is the unique positive number such that 4^2=16.

We denote:

16^{1/2}=4

The number -4 also satisfies (-4)^2=16.

The equation has two solutions : 4 and -4.

### Monomial equation

A monomial equation is an equation of the form:

**ax^n=b**

a and b are real numbers.

Consider the following monomial equation:

3x^2=12

The monomial equation is solved for by dividing by 3 and then applying the previous rules from above:

3x^2=12

x^2=4

x=2 or x=-2

Therefore the monomial equation has two solutions:

- x=2
- x=-2

Consider the following monomial equation:

-3x^2=12

It has no real solutions. This is shown as follows:

-3x^2=12

x^2=-4

The equation has no solution because no real number raised to an even power is negative.