What is the difference between rational exponents and radicals?

Answer 1

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:

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:

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...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, 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".

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 , wherepi 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:

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

(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:

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:

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:

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 "4".

"Roots" (or "radicals") are the "opposite" operation of applying exponents; you can "undo" a power with a radical, and a radical can "undo" a power. For instance, if you square 2, you get 4, and if you "take the square root of 4", you get 2; if you square 3, you get 9, and if you "take the square root of 9", you get 3: Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved

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The "" symbol is called the "radical"symbol. (Technically, just the "check mark" part of the symbol is the radical; the line across the top is called the "vinculum".) The expression " " is read as "root nine", "radical nine", or "the square root of nine".

You can raise numbers to powers other than just 2; you can cube things, raise them to the fourth power, raise them to the 100th power, and so forth. In the same way, you can take the cube root of a number, the fourth root, the 100th root, and so forth. To indicate some root other than a square root, you use the same radical symbol, but you insert a number into the radical, tucking it into the "check mark" part. For instance:

The "3" in the above is the "index" of the radical; the "64" is "the argument of the radical", also called "the radicand". Since most radicals you see are square roots, the index is not included on square roots. While " " would be technically correct, I've never seen it used.

a square (second) root is written as

a cube (third) root is written as

a fourth root is written as

a fifth root is written as:

You can take any counting number, square it, and end up with a nice neat number. But the process doesn't always work going backwards. For instance, consider , the square root of three. There is no nice neat number that squares to 3, so cannot be simplified as a nice whole number. You can deal with in either of two ways: If you are doing a word problem and are trying to find, say, the rate of speed, then you would grab your calculator and find the decimal approximation of :

Then you'd round the above value to an appropriate number of decimal places and use a real-world unit or label, like "1.7 ft/sec". On the other hand, you may be solving a plain old math exercise, something with no "practical" application. Then they would almost certainly want the "exact" value, so you'd give your answer as being simply "".

Simplifying Square-Root Terms

To simplify a square root, you "take out" anything that is a "perfect square"; that is, you take out front anything that has two copies of the same factor:

Note that the value of the simplified radical is positive. While either of +2 and -2 might have been squared to get 4, "the square root of four" is defined to be only the positive option, +2. When you solve the equation x2 = 4, you are trying to find all possible values that might have been squared to get 4. But when you are just simplifying the expression , the ONLY answer is "2"; this positive result is called the "principal" root. (Other roots, such as -2, can be defined using graduate-school topics like "complex analysis" and "branch functions", but you won't need that for years, if ever.)

Sometimes the argument of a radical is not a perfect square, but it may "contain" a square amongst its factors. To simplify, you need to factor the argument and "take out" anything that is a square; you find anything you've got a pair of inside the radical, and you move it out front. To do this, you use the fact that you can switch between the multiplication of roots and the root of a multiplication. In other words, radicals can be manipulated similarly to powers:

• Simplify

• There are various ways I can approach this simplification. One would be by factoring and then taking two different square roots:

The square root of 144 is 12.

You probably already knew that 122 = 144, so obviously the square root of 144 must be 12. But my steps above show how you can switch back and forth between the different formats (multiplication inside one radical, versus multiplication of two radicals) to help in the simplification process.

• Simplify

• Neither of 24 and 6 is a square, but what happens if I multiply them inside one radical?

• Simplify

This answer is pronounced as "five, root three". It is proper form to put the radical at the end of the expression. Not only is "" non-standard, it is very hard to read, especially when hand-written. And write neatly, because "" is not the same as "".

You don't have to factor the radicand all the way down to prime numbers when simplifying. As soon as you see a pair of factors or a perfect square, you've gone far enough.

• Simplify

• Since 72 factors as 2×36, and since 36 is a perfect square, then:

Since there had been only one copy of the factor 2 in the factorization 2×6×6, that left-over 2 couldn't come out of the radical and had to be left behind.

• Simplify