Monomials are exponential expressions with non-negative integer exponents. There are three parts to a monomial: the base, the coefficient, and the exponent. The base is repeatedly multiplied by itself according to the exponent.
Consider the example of a monomial shown here:
In this example, the base is x. The exponent is 3, and indicates how many times the base is used in repeated multiplication. The coefficient is 5, which is the number that acts as a multiplier to the base and the exponent.
You can multiply monomials with and without coefficients. Consider this example of an expression without coefficients:
To simplify, you can expand the expression using repeated multiplication. Therefore, x squared becomes x times x, and x to the fourth becomes x times x times x times x. Multiplying the x terms together provides x to the sixth.
In the expression above, you can also determine the exponent of the product by adding the two exponents, 2 plus 4, together to provide x to the sixth. This illustrates the product property for exponential expressions, which works for the product of all monomials when the bases of the monomials are the same. In general, this states that x^a times x^b is equal to x^(a+b).
This next example involves multiplying monomials with coefficients:
You can use the commutative property of multiplication, which allows you to group your coefficients and variables together. Multiplication is commutative because you can multiply numbers in any order. Therefore, grouping your coefficients and your x terms together gives you:
Multiplying -3 times 5 equals -15, and multiplying x times x^6 is the same as x^1 times x^6. Adding your exponents using the product property for exponents gives you x^7, providing your final solution of:
Exponents are generally integers or fractions, but they may be any number, such as a decimal. A rational exponent is an exponent that can be represented as a fraction.
The product property applies to all types of exponents, including integers and fractions. Therefore, you add fractions when applying the product property of exponents to rational exponents. You can easily add fractions when they have common denominators:
You need to add your exponents, meaning that you need to add the fractions together. Your denominators are already the same, so you can simply add your numerators together.
Now, both your numerator and denominator have a common factor of 2, so you can cancel it out by dividing them both by 2, which gives you 3 over 1, or 3.
Bringing back in the product of your coefficients, your final expression becomes:
Source: This work is adapted from Sophia author Colleen Atakpu.
An exponential expression with non-negative integer exponents.