What Do Fractional Exponents Mean?

Depending on the person – Fractions are either fun or downright terrifying.

When fractions show up in your exponents, that’s when the real nightmare begins. Or does it?

At Cleverism we like to explain things in a fun and interesting way. Even if you aren’t a math whiz, we’ll provide a modest explanation that allows you to understand what fractional exponents are.

Let’s get down to it.

A BRIEF OVERVIEW OF EXPONENTS

Before we explain fractional exponents, let’s get a quick math lesson on what exponents are. If you already understand what exponents are, you can skip this and head directly to the fractional exponent’s section below.

In brief, an exponent is when a number is multiplied by its own number a specific number of times.

For example, 6 x 6 x 6 = 216

In exponential form, the number is written as the following – 6³

Let’s take another example.

2⁴  is broken down into 2 x 2 x 2 x 2 = 16.

So effectively 2⁴  = 16.

Where 2 is the base number and 4 is the exponent while 16 is the sum of it.

Exponents are a convenient way to write down an otherwise tedious way of multiplying numbers.

Take writing 2⁹ for example. In non-exponential form that’s 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2.

Notice how complicated it gets to read and write when you don’t use an exponent?

Exponents are also known as Indices and Powers. The above example could be known as “2 to the power of 9” or “2 to the 9^th power”.

Another symbol that is used to denote an exponent is ^.

So, 2⁹ is written as 2^⁹ sometimes to denote an exponent. This symbol is found on the numerical ‘6’ on

Now you’re probably left wondering what if the exponent is 1 such as 7¹, well that’s an easy one. The answer is 7. Any exponent that is 1 is the base number itself.

However, if the exponent is 0, the answer is always 1. So, 7⁰ = 1.

There, simple isn’t it?

Before we move on to fractional exponents, there’s one other critical aspect of exponents. Negative Exponents

Negative Exponents

A negative exponent is the inverse of an exponent. Instead of multiplying, we do the opposite – Divide.

Here’s an example.

Let’s find the answer to xⁿ.

X = 2

N = -3

Therefore, a negative exponent is written as 2^-3.

Now let’s break it down into a workable format.

Answer = 0.125

And that’s all there’s to it. Calculating a negative exponent is directly opposite to working with an exponent.

Once you understand exponents and negative exponents, fractional exponents are easier to understand.

Let’s move on to the main topic – Fractional Exponents.

WHAT ARE FRACTIONAL EXPONENTS?

Fractional exponents are simple than they appear. They are used in core algebraic expressions to streamline mathematical equations.

Fractional exponents are also commonly used over radical signs which are denoted by ‘ √ ‘. Fractional exponents are commonly used when calculating square roots.

In the previous section, we learned about exponents such as 4² or 5⁹ or 9³.

Examples of fractional exponents are  4^2/5 or 5^4/5 or 9^6/4. Fractional exponents are also written as. With ‘x’ being the base, ‘n’ denoting the numerator and ‘d’ being the denominator.

Notice the numerator and denominator appearing with the base number.

To better understand how to solve fractional exponents, let’s perform a simple example.

4^1/2 =?

In the above fractional exponent 4 = base number. 1 is the numerator and 2 is the denominator.

The workable format to find the solution would be

Since 4 to the power of 1 is 4

Therefore,

Hence the principal root of

 is 2.

The answer is 2.

Let’s do another example to ensure you understand.

Example 2

8^2/3 =?

First, we break the fraction into parts.

We can write 8^2/3  as  [8^1/3]²

Now we find the cube root of 8.

A cube root will always multiply itself by thrice to reach the base number.

In this case, the cube root of 8 is 2 as 2 x 2 x 2 = 8.

So, the next part is [2]². Which is 2 x 2 = 4.

The answer to 8^2/3 = 4

Another method to find out the solution is by converting 8^2/3 into.

8²  =  8 x 8 =  64.

So,

The cube root of 64 = 4 since 4 x 4 x 4 = 64.

The answer is 4.

Two different methods to find the solution, use the one that suits you best.

Here’s a video guide explaining fractional exponents in a simple way.

THE LAW OF EXPONENTS

Much like many mathematical expressions, there are exponent rules required to function. Understanding these laws creates a convenient setting for figuring out how exponents work.

The exponent laws are also known as the law of indices.

1. The Multiplication (Bases) Law

When the bases of the multiplication match such as x^a x^b, the result is x^a+b.

2. The Multiplication (Powers) Law

When the powers of the multiplication match such as x^a y^a_{, the result is} (xy)^a

3. The Division (Bases) Law

When the bases of the division match such as x¹ / x² , the result is x^1-2

4. The Division (Powers) Law

When the powers of the division match such as x¹ / y¹, ^{the result is} (x / y)¹

5. The Powers Law

The following exponent (y^a)^b is also said to be y^ab

6. The Undefined Law

0⁰  is considered a zero exponent and can come out as either 0 or 1. The answer is usually said to be ‘indeterminate’ or ‘undefined’.

LIST OF ONLINE MATH RESOURCES & TOOLS

Here is a list of useful tools that aid you in calculating your mathematical equations.

1. Fraction Calculator

An online calculator to find out simple and mixed fractions. Comes with a ‘Reset’ option.

2. Exponent Calculator

Enter the base and exponent and the answer pops up. As simple as it gets.

3. Square Root Calculator

Enter a number to get the square root.

4. Cube Root Calculator

Enter a number to get the cube root.

5. Power Mod Calculator

A calculator dedicated to Power mods.

6. Exponent Tables & Patterns

A quick view of tables of exponents. Great for convenient reference when performing your first few fractional exponents.

7. Order Tables from 1-12

A power table for advanced fractional exponent users from 1-12. Comes with user-friendly printable access.

CONCLUSION

At first, tackling fractional exponents seems confusing. With the simplistic steps mentioned in this write-up, even a novice math user can calculate equations. Remember, every rule that applies to exponents directly applies to fractional exponents as well.

Today, fractional exponents are used in wide variety of jobs such as

• Computer Programming
• Accountants
• Geologists
• Chemical Engineers
• Financial Advisor
• Aerospace Engineer

And many more. It’s critical to up your math game and perfect your exponents to impress interviewers.

Love or hate fractional exponents? Share your thoughts in the comments below.