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Welcome to our Math lesson on **The Definition of Monomials**, this is the first lesson of our suite of math lessons covering the topic of **The Definition of Monomials and Polynomials**, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

In Chapter 6, we briefly mentioned the concept of monomers and polynomials. Thus, a monomial is an algebraic expression consisting of the product of a real number called a coefficient and one or more letters representing variables, which are raised to certain natural powers. For example,

2x^{5}; *1**/**2* x^{3} y; -4ab^{5}; etc.

are all monomials, as all variables are preceded by a number (coefficient), and moreover, the variables are raised to positive integer powers.

On the other hand,

are not monomials, as their variables are not always positive integers. Indeed, from the properties of indices and roots (more specifically, ** 1/x**n = x - n and √x = x

2√x = 2 ∙ x

and

Since the index of x in the second expression and that of y in the first and third expressions are not positive integers (y has a negative integer index, while x has a rational index), they are not monomials.

Which of the following algebraic expressions is a monomial?

x*2**/**3*^{4}yz^{5}- 5x
^{1/3}yz^{3} ab*-4**/**3*^{7}- -
*3a√b**/**b*^{1/2}

- The expression is a monomial, because it contains a rational (and therefore real) number (
x*2**/**3*^{4}yz^{5}) as a coefficient, which precedes three variables: x, y and z, raised to the fourth, first and fifth power respectively (i.e. all variables are raised in natural powers).*2**/**3* - The expression 5xis not a monomial, because one of the variables, (variable x) is rational (therefore not natural), as its index is
^{1/3}yz^{3}.*1**/**3* - The expression is a monomial, because it contains a rational (and therefore real) number (
ab*-4**/**3*^{7}) as a coefficient, which precedes two variables: a and b, raised to the first and seventh power respectively (i.e. all variables are raised in natural powers).*-4**/**3* - Apparently, the expression is not a monomial as the indices of its variable b are not natural. However, by means of a few transformations (given the condition that b must not be zero), we obtain a monomial, because
*-(3a√b)**/**b*^{1/2}Hence, variable b is simplified.=*-(3a√b)**/**b*^{1/2}= -3a*-(3ab*^{1/2})*/**b*^{1/2}

Enjoy the "The Definition of Monomials" math lesson? People who liked the "The Definition of Monomials and Polynomials lesson found the following resources useful:

- Definition Of Monomials Feedback. Helps other - Leave a rating for this definition of monomials (see below)
- Polynomials Math tutorial: The Definition of Monomials and Polynomials. Read the The Definition of Monomials and Polynomials math tutorial and build your math knowledge of Polynomials
- Polynomials Revision Notes: The Definition of Monomials and Polynomials. Print the notes so you can revise the key points covered in the math tutorial for The Definition of Monomials and Polynomials
- Polynomials Practice Questions: The Definition of Monomials and Polynomials. Test and improve your knowledge of The Definition of Monomials and Polynomials with example questins and answers
- Check your calculations for Polynomials questions with our excellent Polynomials calculators which contain full equations and calculations clearly displayed line by line. See the Polynomials Calculators by iCalculator™ below.
- Continuing learning polynomials - read our next math tutorial: Operations with Polynomials

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