Rational Exponents: Rules & Definition
- Rational exponents are a notation used to show a number or variable (called the base) is being raised to a power and/or having a root taken. The top number in the rational exponent symbolizes the exponent while the bottom number symbolizes the radical. For instance, X^(3/2) can also be written as sqrt(x^3).
- Both radical form and rational exponent form are useful for simplifying algebraic expressions, but sometimes one will be faster and efficient than the other. For instance, rational exponents make it easy to simplify expressions involving multiplication and division because you can use exponent rules on them.
When multiplying like bases, add their exponents and, when dividing like bases, subtract their exponents.
For instance, you can simplify sqrt(X^3) x sqrt(X^5) x 6thrt(X^7) by changing it to X^(3/2) x X^(5/2) x X^(7/6). Use the common denominator to get X^(31/6). - As with normal exponents, rational exponents that are being raised to a second exponent are multiplied by that exponent. For example, (R^(5/2))^3 becomes R^(15/2)
- Although rational exponents may be more useful in some circumstances, and although problems are sometimes written with rational exponents, answers should be expressed in radical form. Always re-check radical answers to see if they can be simplified further. For example, the answer in Section 2 would convert to 6thrt(X^31).
- Negative exponents indicate reciprocals. For example, X^-3 means 1/(x^3). Radicals are not allowed to remain in the denominator of a fraction. That means if an expression contains a negative rational exponent, it is not simplified, because there would be a radical in the denominator if it were written in radical form. To simplify, rewrite it in radical form. Simplify the radical and then multiply both the numerator and denominator by the radical in the bottom.
For instance, 2Z^(-1/2) would become 2/sqrt(Z), which you would multiply by sqrt(Z)/sqrt(Z) to get 2sqrt(Z)/Z.