How To Factor With Negative Fractional Exponents

A positive exponent tells you how many times to multiply the base number by itself. For example, the exponential term ​y​³ is the same as ​y​ × ​y​ × ​y​, or ​y​ multiplied by itself twice. Once you've grasped that basic concept, you can start to add on extra layers like negative exponents, fractional exponents or even a combination of both.

Before factoring negative, fractional exponents, let's take a quick look at how to factor negative exponents, or negative powers, in general. A negative exponent does exactly the inverse of a positive exponent. So while a positive exponent like ​a​⁴ tells you to multiply ​a​ by itself three times (so there are four in total in the expression), or ​a​ × ​a​ × ​a​ × ​a,​ seeing a negative exponent tells you to ​divide​ by ​a​ four times: so

\(a^{-4} = \frac{1}{a × a × a × a}\)

Or, to put it more formally:

\(x^{-y} = \frac{1}{x^y}\)

The next step is learning how to factor fractional exponents. Let's start with a very simple fractional exponent, such as ​x​^1/​y​. When you see a fractional exponent like this, it means you must take the ​y​th root of the base number. To put it more formally:

\(x^{1/y} = \sqrt[y]{x}\)

If that seems confusing, a few more concrete examples can help:

\(y^{1/3} = \sqrt[3]{y}\)
\(b^{1/2 }= \sqrt{b}\)

(Remember, √​x​​ ​is the same as ²√​x;​ but this expression is so common that the ², or index number, is omitted.)

\(8^{1/3} = \sqrt[3]{8 }= 2\)

What if the numerator of the fractional exponent isn't 1? Then that number's value remains as an exponent, applied to the entire "root" term. In formal terms, that means:

\(y^{m/n} = (\sqrt[n]{y})^m\)

As a more concrete example, consider this:

\(a^{b/5} = (\sqrt[5]{a})^b\)

When it comes to factoring negative fractional exponents, you can combine what you've learned about factoring expressions with negative exponents and those with fractional exponents.

Remember,

\(x^{-y} = \frac{1}{x^y}\)

regardless of what's in the ​y​ spot; ​y​ could even be a fraction.

So if you have an expression ​x​ ^{−​a​/​b​}, that's equal to 1/(​_x^a_​^/​b​). But you can simplify a step further by also applying what you know about fractional exponents to the term in the denominator of the fraction.

Remember,

\(y^{m/n} = (\sqrt[n]{y})^m\)

or, to use the variables you're already dealing with,

\(x^{a/b} = (\sqrt[b]{x})^a\)

So, going that further step in simplifying ​x​ ^{−​a​/​b​}, you have

\(x^{-a/b} = \frac{1}{x^{a/b}} = \frac{1}{(\sqrt[b]{x})^a}\)

That's as far as you can simplify without knowing more about ​x​, ​b​ or ​a.​ But if you do know more about any of those terms, you might be able to simplify further.

To illustrate that, here's one more example with a bit more information added:

Simplify

\(16^{-4/8}\)

First, did you notice that −4/8 can be reduced to −1/2? So you have 16 ^−1/2, which already looks a lot friendlier (and maybe even more familiar) than the original problem.

Simplifying as before, you'll arrive at

\(16^{-1/2} = \frac{1}{(\sqrt[2]{16})^1}\)

which is usually written simply as

\(\frac{1}{\sqrt{16}}\)

And since you know (or can quickly calculate) that √16 = 4, you can simplify that one last step to:

\(16^{-4/8} = \frac{1}{4}\)