Fractional (Rational) Exponents

For the square (or "second") root, we can write it as the one-half power, like this:

...or:

The cube (or "third") root is the one-third power:

The fourth root is the one-fourth power:

The fifth root is the one-fifth power; and so on.

Looking at the first examples above, 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, say, 15^(4/5), you must put parentheses around the "4/5", because otherwise your calculator will think you mean "(15 ⁴) ÷ 5".

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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 "3^5.5", recall that 5.5 = 11/2, so:

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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, 3π, where π is the number you learned about in geometry, and is approximately equal to 3.14159, cannot be simplified or rearranged as a radical.

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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 the number –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:

So if they give you, say, x^3/6, 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 x^4/6, 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 x^4/5, then you don't have to 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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A technological point: Calculators and other software do not compute things the way people do; they use pre-programmed algorithms. Sometimes the particular method the calculator uses can create difficulties in the context of fractional exponents.

For instance, you know that the cube root of –8 is –2, and the square of –2 is 4, so (–8)^(2/3) = 4. But some calculators return a complex value or an error message, as is the case with one of my graphing calculators:

Clearly, this isn't the expected result, especially if you haven't yet studied complex numbers. This calculator answer isn't helpful.

If you enter "=(–8)^(2/3)" into a cell, the Microsoft "Excel" spreadsheet returns the error "#NUM!", another unhelpful answer.

Some calculators and programs will do the computations as expected, as displayed at right from my other graphing calculator:

The difference has to do with the pre-programmed calculating algorithms. These algorithms generally try to do the computations in ways which require the fewest "operations", in order to process what you've entered as quickly as possible.

But sometimes the fastest method isn't always the most useful, and your calculator will "choke".

Fortunately, you can get around the problem. By splitting the numerator and denominator of the fractional power, you can enter the expression so that your calculator should arrive at the correct value. After getting the unhelpful answer in my first calculator, I re-entered the number with the power broken into pieces:

As you can see above, it didn't matter if I first took the cube root of negative eight and then squared, or if I first squared and then cube-rooted; either way, by feeding the numerator and denominator to the calculator separately, I was able to get the calculator to return the proper value of "4".