Overview
Rational exponents are an extension of the rules of exponents. They can be solved by applying those rules and often take more than one step.
What Is an Exponent?
An exponent such as 32 is a direction to multiply 3 by itself twice, such as 3∙3, or 9. Similarly, an exponent such as 23 is a direction to multiply 2 by itself 3 times, as 2∙2∙2. It is not the same as 2∙3, which is 6. Therefore, a number such as an means to multiply a by itself n times. The rules of exponents have been extended so that x0 equals 1, so that 50 equals 1, 70 equals 1, and so forth. By definition, x1is x by itself, so that 21 equals 2, 51 equals 5, and 100,521,3621 equals 100,521,362.
What about Roots of Numbers?
By definition, the root of a number is the inverse operation of multiplying by an exponent. If 32 equals 9, the square root of 9 (√9) is 3. Similarly, if 53 means 5∙5∙5, then the cube root of 125 is 5. Also, if 24 equals 16, then the fourth root of 16 is 2. In addition, (√2)(√2) is the same thing as saying (√2)2 or 2. That is also the same thing as 21.
What about Rational Exponents?
By the rules of adding exponents, 22∙∙24 is the same thing as 26, or 22+4. Therefore, 2½ ∙ 2½would be the same as 2½ + ½ or 21. Now 21 equals 2, and so does (√2)2. By definition, the square root of any positive number n, when n>0, equals n1/2. That definition can also be extended to roots other than square roots, such that n1/3 means the cube root of n. Therefore, 17281/3is equal to the cube root of 1728, which is 12, because 123 is 1728.
What about Other Fractional Exponents?
The rules of defining fractional exponents such as 251/2 mean that 251/2 is the same thing as the square root of 25, or 5. A fractional exponent as 253/2, however, means (25 1/2) 3 or (√25)3. It is a good rule to take the square root first, so the square root of 25 is 5, and 53is 125. Suppose the number were –(16)1/2. The square root of 16 is 4, and taking the negative would be -4. The square root of -16 is not a real number.
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