Radicals and Rational Exponents

Having discussed the topic of radicals and nth roots in my previous set of posts, we can apply that information to the concept of fractional exponents. These rational exponents are no more difficult to work with, and once you identify just what they are telling you, you can easily apply your rules for exponents to simplify them and solve your equations. Factoring fractional exponents may look tricky, but you will see that they follow the same rules that you already know.

Consider the expression 2^1/3. Knowing your rules for exponents, you can see that if we cube this so that we get (2^1/3)³, this is equal to 2¹, which is just 2. However, what you will realize is that what we have just done is apply property #1 of the properties of nth roots. That is, (ⁿ√x)ⁿ = x. So, what we have just shown is that 2^1/3 is the same as ³√2.

So then, with that example we can now extend what we've done to come to a definition of rational exponents and roots.

Part 1:  If we let b be any real number, and n is a natural number, b^1/n can be defined by ⁿ√b. (if n is even, b must be greater than or equal to 0).

Furthermore, we can extend our definition of exponents fractions to include any fraction, not just 1/x type.

Part 2:  If we let m/n be a rational number, n is positive, and ⁿ√b exists, we can say that b^m/n = ⁿ√(b)^m or (ⁿ√b)^m. (Both forms say the same thing and work the same way.)

It is important to realize and understand that these rational exponents follow the same properties as integer exponents, which you are likely already familiar with (I'll do a post soon to refresh you, or for those who have not learned that topic yet).

So now that you know what rational exponents are, let's take a look at a few examples of how you work with radical and rational exponents.  Simplifying rational exponents takes some practice, but it is all the same math that you have already learned.

1.  Simplify the following expressions:
a) 64^1/2
b) -64^1/2
c) (-64)^1/2
d) 64^-1/2

a) This is simply the square root of 64, √64, which is 8.
b) Recall the notation with negative signs and nth roots. This means -(64^1/2) = -√64 = -8.
c) As there is no such real number that gives -64 when squared, this expression cannot be simplified to a real number.
d) 64^-1/2 indicates √64^-1, and a negative exponent means that you take the inverse of it. So,64^-1 means 1/64, and then the square root of this is 1/8.

2.  Simplify (4x^2/3)(3x^1/4)
(4x^2/3)(3x^1/4) = 12x^{(2/3) + (1/4)} (by the exponent rules)
= 12x^11/12

3.  Simplify (√x)(³√y⁴)
= (x^1/2)(y^3/4)
Alternatively, if we give the fractions a common denominator, we can rewrite as follows:
(x^1/2)(y^3/4) = (x^2/4)(y^3/4) = (⁴√x²)(⁴√y³) = ⁴√(x²y³)

So, there you have it.  Working with radicals and rational exponents is not that different from working with any other exponents, once you know what you're doing.  Go over my examples again to be sure you understand the concepts, and practice some more on your own, and you will be an expert in no time. Drop me a line in the comments if you have any problems.

One final note: irrational or radical exponents follow the same rules as well, and I will cover some of those examples in a future post.

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