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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:

    cbrt(2^3) = 2,  4th-rt(3^4) = 3   Copyright Elizabeth Stapel 2000-2011 All Rights Reserved

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 the one-half power, like this:




    sqrt(2) = 2^(1/2)


    sqrt(4) = 4^(1/2) = 2

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

    cbrt(8) = 8^(1/3) = 2

The fourth root is the one-fourth power:

    4th-rt(81) = 81^(1/4) = 3

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

Looking at the first examples, we can re-write them like this:

    cbrt(2^3) = (2^3)^(1/3) = 2^((3/1)(1/3)) = 2^1 = 2, 4th-rt(3^4) = (3^4)^(1/4) = 3^((4/1)(1/4)) = 3^1 = 3

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".

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:

    (10th-rt(25))^5 = (25^(1/10))^5 = 25^((1/10)(5/1)) = 25^(1/2) = sqrt(25) = 5

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:

    7^(2/3) = cbrt(7^2) = (cbrt(7))^2

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 , where pi is the number approximately equal to 3.14159, 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 the number 2. Then:

    sqrt[ (-2)^2 ] = sqrt(4) = 2, does not equal -2

In other words, you put in a negative number, and got out a positive number! This is the official definition of absolute value:

    | x | = sqrt(x^2)

(Yeah, I know: they never told you this, but they expect you to know somehow, 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 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".)

A technology 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:

(-8)^(2/3) returns the complex value (-2, 3.46410161514)

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

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

(-8)^(2/3) returns the expected value of 4

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, your calculator should arrive at the correct value:

((-8)^(1/3))^2 and ((-8)^2)^(1/3) each return the expected value of 4

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, the calculator returned the proper value of "

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Cite this article as:

Stapel, Elizabeth. "Fractional (Rational) Exponents." Purplemath. Available from Accessed


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