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Fractional (Rational) Exponents (page 5 of 5) Sections: Basics, Negative exponents, Scientific notation, Engineering notation, Fractional exponents You already know of one relationship between exponents and radicals: the appropriate radical will "undo" an exponent, and the right power will "undo" a root. For example:
But there is another relationship (which, by the way, can make computations like those above much simpler). For the square (or "second") root, we can write it as a power, like this:
...or: Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved
The cube (or "third") root is also the one-third power:
The fourth root is also the one-fourth power:
The fifth root is also the one-fifth power; and so on. Looking at the first examples, we can re-write them like this:
You can enter fractional exponents on your calculator for evaluation, but you must remember to use parentheses. If you are trying to evaluate, for instance, 15(4/5), you must put parentheses around the "4/5", because otherwise your calculator will think you mean "(15 4) ÷ 5". Fractional exponents allow greater flexibility (you'll see this a lot in calculus), are often easier to write than the equivalent radical format, and permit you to do calculations that you couldn't before. For instance:
Whenever you see a fractional exponent, remember that the top number is the power, and the lower number is the root (if you're converting back to the radical format). For instance:
By the way, some decimal powers can be written as fractional exponents, too. If you are given something like "35.5", recall that 5.5 = 11/2, so: 35.5 = 311/2 Generally, though, when you get a decimal power (something other than a fraction or a whole number), you should just leave it as it is, or, if necessary, evaluate it in your calculator. For instance, 3 pi cannot be simplified or rearranged as a radical. A technical point: When you are dealing with these exponents with variables, you might have to take account of the fact that you are sometimes taking even roots. Think about it: Suppose you start with –2. Then:
In other words, you put in a negative number, and got out a positive number! This is the official definition of absolute value:
(Yeah, I know: they never told you this, but they expect you to know, so I'm telling you now.) So if they give you, say, x3/6, then x had better not be negative, because x3 would still be negative, and you would be trying to take the sixth root of a negative number. If they give you x4/6, then a negative x becomes positive (because of the fourth power) and is then sixth-rooted, so it becomes | x |2/3 (by reducing the fractional power). On the other hand, if they give you something like x4/5, then you don't care whether x is positive or negative, because a fifth-root doesn't have any problem with negatives. (By the way, these considerations are irrelevant if your book specifies that you should "assume all variables are non-negative".)
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![sqrt[ (-2)^2 ] = sqrt(4) = 2, does not equal -2](exponent/exp23.gif){kind=small}
