Factoring 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.

TL;DR (Too Long; Didn’t Read)

A negative, fractional exponent ​_y_​ −​_m_​/​_n_​ can be factored to the form:

1 / (​_n_​√​_y_​)​_m_​

Factoring Negative Powers

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})

Factoring Fractional Exponents

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)

Combining Negative and Fractional Exponents

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/(​_xa_​/​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.

Another Example of Simplifying Fractional Negative Exponents

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})