The Similarities & Differences Between Rational Expressions & Rational Number Exponents

Rational expressions and rational exponents are both basic mathematical constructs used in a variety of situations. Both types of expressions can be represented both graphically and symbolically. The most general similarity between the two is their forms. A rational expression and a rational exponent are both in the form of a fraction. Their most general difference is that a rational expression is composed of a polynomial numerator and denominator. A rational exponent can be a rational expression or a constant fraction.

A rational expression is a fraction where at least one term is a polynomial of the form ax² + bx + c, where a, b and c are constant coefficients. In the sciences, rational expressions are used as simplified models of complex equations in order to more easily approximate results without requiring time-consuming complex math. Rational expressions are commonly used to describe phenomena in sound design, photography, aerodynamics, chemistry and physics. Unlike rational exponents, a rational expression is an entire expression, not just a component.

The graphs of most rational expressions are discontinuous, meaning they contain a vertical asymptote at certain values of x that are not part of the domain of the expression. This effectively splits the graph up into one or more sections, divided by the asymptote. These discontinuities are caused by values of x that lead to division by zero. For example, for the rational expression 1 / (x – 1)(x + 2), discontinuities are located at 1 and -2 since at these values the denominator equates to zero.

An expression with a rational exponent is simply a term raised to the power of a fraction. Terms with rational number exponents are equivalent to root expressions with the degree of the denominator of the exponent. For example, the cube root of 3 is equivalent to 3^(1/3). The numerator of the rational exponent is equivalent to the power of the base number when in its radical form. For example, 5^(4/5) is equivalent to the fifth root of 5^4. A negative rational exponent indicates the reciprocal of the radical form. For example, 5^(-4/5) = 1 / 5^(4/5).

Graphs with rational exponents are continuous everywhere except for the point x / 0, where x is any real number, since division by zero is undefined. The graphs of terms with rational exponents are horizontal lines because the value of the expression is constant. For example, 7^(1/2) = sqrt(7) never changes values. Unlike rational expressions, graphs of terms with rational exponents are always continuous.