Is this a true statement?
Polynomials and radical expressions differ because radical expressions contain rational numbers as exponents and polynomials do not.
I think you have it backwards, with some additonal miconceptions.
Polynomials have rational (and, in fact, integer) exponents.
Radical expressions contain variables to fractional powers. A square root corresponds to an exponent of 1/2. That is still a rational number.
To determine if this statement is true, we need to examine the definitions and characteristics of polynomials and radical expressions.
First, let's clarify what polynomials and radical expressions are:
- Polynomial: A polynomial is an algebraic expression consisting of variables, coefficients, and mathematical operations such as addition, subtraction, multiplication, and non-negative integer exponents.
- Radical Expression: A radical expression includes a radical symbol (√) and can involve rational numbers as exponents.
Looking at the statement, it says that radical expressions contain rational numbers as exponents, whereas polynomials do not. This statement is incorrect.
In polynomials, the exponents of the variables must be non-negative integers. For example, x^2, x^3, 3x^4, etc., are valid terms in a polynomial expression.
In radical expressions, the exponent can indeed be a rational number. For instance, √4 is equal to 2, where the fraction 1/2 represents the rational exponent.
To summarize, polynomials do not contain radical symbols and their exponents are non-negative integers, whereas radical expressions can contain rational number exponents.
Therefore, the statement is false as it incorrectly asserts that radical expressions contain rational number exponents while polynomials do not.