Fractional Exponents

Fractional Exponents are used to describe numbers with fractional powers and are also known as Rational Exponents. As any exponent shows how many times a number has been multiplied i.e., 3² = 3 × 3 = 9, but in the case of fractional exponents, it can’t be the case as we can’t multiply 3 to itself 1.5 times. Thus, fractional exponents are natural extensions of integral exponents and are used to calculate the values of fractional powers as well as radicals.

This article provides a well-rounded description of Fractional Exponents, including subtopics such as examples, representation, and laws. Other than that, all the subtopics such as various operations performed with fractional exponents, solving any fractional exponent, negative fractional exponent, and many many more, are discussed too.

What are Fractional Exponents?

Powers and roots can be expressed jointly using fractional exponents. In any general exponential expression of the form ab, a is the base and b is the exponent. When b is given in the fractional form, it is known as a fractional exponent.

Fractional exponent also named as rational exponent which are the expressions that are rational numbers rather than integers. It is an alternate representation for expressing powers and roots together. The general form of fraction exponent is

In a fractional exponent, the numerator is the power and the denominator is the root. In the above example, ‘a’ and ‘b’ are positive real numbers, and x is a real number, a is the power and b is the root.

Examples of Fractional Exponents

Some of examples of fractional exponents are 5^1/2, 8^2/3, etc. So, we can say that the general form of a fractional exponent is x^a/b, where x is the base and a/b is the exponent.

Representation of Fractional Exponent

Fractional Exponent can be represented as:

x^1/a = and for x^a/b = 

Example: Represent 4^1/3

Solution:

x^1/a = 

⇒ 4^1/3 = 

Read more about Fractions.

Fractional Exponents vs Integer Exponents

The following table shows the difference between fractional and integer exponents:

Fractional Exponents	Integer Exponents
Used when the power isis not whole numbers.	Used when the power are whole numbers.
They are written in the form x a/b.	They are written in the form x a
Allows us to express roots and other non- integer powers.	Positive integer exponents indicate repeated multiplication and negative integer exponents indicate repeated division.
Ex: 41/2=2	Ex: 42 = 16 and 4-2 = 1/16

Common Fractional Exponents

Lets find out some common exponents which are used in various places and can be named accordingly as shown in the table below:

Exponent	Name of Exponent	Representation
1/2	Square root	a1/2=
1/3	Cube root	a1/3=
1/4	Fourth root	a1/4 =

Fractions with Fractional Exponents

When we encounter fractional exponents in a fraction then various amazing result arises.

For example: (1/25)^1/2 , here a fractional exponent that is 1/2 is present over a fraction that is 1/25.

So to find this we need to take the square root of both the numerator and denominator. So, (1)^1/2/(25)^1/2 = 1/5

In this way we get the values for fractions with fractional exponents.

Read More,

• Laws of Exponents
• Logarithms
• Laws of Logarithms

How to Solve Fractional Exponents?

To solve fractional exponents we should follow some rules. We can easily multiply or divide numbers with fractional exponents by following to a few instructions. Despite the fact that many individuals are familiar with whole-number exponents, when it comes to fractional exponents, they frequently make mistakes that can be avoided by adhering to the rules of Fractional Exponents given below.

Laws of Fractional Exponents

There are various rule for Fractional Exponents, some of these are:

• a^1/m × a^1/n = a^{(1/m + 1/n)}
• a^1/m ÷ a^1/n = a^{(1/m – 1/n)}
• a^1/m × b^1/m = (ab)^1/m
• a^1/m ÷ b^1/m = (a ÷ b)^1/m
• a^-m/n = (1/a)^m/n

How to Simplify Fractional Exponents?

The concepts of multiplication and division can be used to understand how to simplify fractional exponents. It involves simplifying the expression or exponent into a more understandable form. For example : 4^1/2 can be reduced to 2.

Example: Solve 16^1/4

Solution:

We know that 16 can be expressed 2x2x2x2.

16 = 2⁴

So, we get, (2⁴)^1/4 = 2

The product of the exponents gives 4×1/4=1. ∴ 4√16=16^1/4=2.

Multiplying Fractional Exponents

If same base is given then we have to add the exponents and write the sum on the common base.

Exponents with the same base can be multiplied using the general formula a^1/mx a^1/n = a^{(1/m + 1/n)}.

For example To multiply 3^1/3 and 3^3/4 we need to add the exponents.

Sum of exponents = 1/3 +3/4 = (4+9)/12 = 13/12

So, 3^1/3 x 3^3/4 = 3^13/12

Example: Solve 2^2/3 * 2^3/4

Solution:

Multiple fractional exponents with same base

Sum of exponents = 2/3 +3/4 = (8+9)/12 = 17/12

So, 2^2/3 x 2^3/4 = 2^17/12

Dividing Fractional Exponents

The division of fractional exponents can be classified into two types

Case 1: Division with Different Powers but the same bases

If same base is given but powers are different then we have to subtract the exponents.

In this case we express it as a^1/m ÷ a^1/n = a^{(1/m – 1/n)}.

For example, 2^3/4 ÷ 2^1/2 = 2^(3/4-1/2), which is equal to 2^1/4.

Example: Simplify 4^3/4 ÷ 4^5/8

Solution:

In this case we express it as a^1/m ÷ a^1/n = a^{(1/m – 1/n)}.

= 4 ^(3/4-5/8) = 4 ^(1/8)

Case 2: Division of fractional exponents with the same powers but different bases

If different base is given but powers are same then we have just divide the base and take the exponent as common.

In this case we express it as a^1/m ÷ b^1/m = (a ÷ b)^1/m.

So we are dividing the bases in the given sequence and writing the common power on it.

For example, 27^3/5 ÷ 3^3/5 = (27/3)^3/5, which is equal to 9^3/5.

Example: Simplify 16^3/5 ÷ 4^3/5

Solution:

Using a^1/m ÷ b^1/m = (a ÷ b)^1/m

So, (16/4)^(3/5) = 4^3/5

Negative Fractional Exponents

Rational exponents are equivalent to negative fractional exponents. In this case, the power has a negative sign and a fractional exponent. For example: 3^-1/2

Applying the exponents principles that state a^-m = 1/a^m is necessary to solve negative exponents. It signifies that the initial step is to raise the reciprocal of the base to the specified power without the negative sign before further simplifying the equation.

Note: a^-m/n = (1/a)^m/n is the general formula for negative fractional exponents.

Example: Simplify 49^-1/2

Solution:

Here the base is 49 and the power is -1/2. The first step is to take the reciprocal of the base, which is 1/49, and remove the negative sign from the power.

We get (1/49)^1/2. As we know that is the square of 7 is 7² = 49, we can re-write the expression as 1/(7²)^1/2. Since 2 and 1/2 cancel each other, so we get 1/7.

Read More,

• Comparing Fractions
• Addition of Fractions
• Subtracting Fractions

Sample Problems on Fractional Exponents

1. Simplify (27/125)^2/3

Solution:

Here both the base and the exponent are in fractional form. 27 can be expressed as a cube of 3 and 125 can be expressed as a cube of 5.

so, 27=3³ and 125=5³.

We get, (3³/5³)^2/3 here3 is a common power for both the numbers, So, ((3/5)³)^2/3, which is equal to (3/5)² as 3×2/3=2.

Now, we have (3/5)², which is equal to 9/25.

2. Simplify 81^-1/4

Solution:

Here the base is 81 and the power is -1/4. The first step is to take the reciprocal of the base, which is 1/81, and remove the negative sign from the power.

We get (1/81)^1/4. As we know that is the fourth power of 3 is 3⁴ = 81, we can re-write the expression as 1/(3⁴)^1/4. Since 4 and 1/4 cancel each other, so we get 1/3.

3. Evaluate 8^1/2 ÷ 2^1/2

Solution:

In this case the powers are the same but the bases are different.

Hence, we can solve this problem as, 8^1/2 ÷ 2^1/2 = (8/2)^1/2 = 4^1/2 = 2.

4. Solve the given expression involving the multiplication of terms with fractional exponents.

3^1/2 × 3^1/4 × 3^1/8

Solution:

The given expression can be re-written as,

3^1/2 × 3^1/4 × 3^1/8

Multiplication of fractional exponents with the same base is done by adding the powers and writing the sum on the common base.

⇒ 3^{(1/2 + 1/4 + 1/8)}

⇒ 3^7/8

Fractional Exponents – Practice Problems

Problem 1: Simplify:

• 16^3/2
• 125^1/3
• 81^3/4
• 64^2/3
• √(x³)

Problem 2: Evaluate:

• 1/8^2/3
• (9^1/2)^3/2
• (16^3/4)^2/3
• (27^2/3)^1/3

Problem 3: Calculate:

• a^2/5 × a^3/5
• 3^3/4 × 3^1/4
• x^1/4 × x^-1/4
• 2^2/5 × 2^-3/5

Fractional Exponents – FAQs

1. What Do Fractional Exponents Mean?

When a number has a fractional exponent, its power is expressed as a fraction rather than an integer. For instance, in the fraction a^m/n, the base is “a” and the power is “m/n.”

2. What is the Rule for Fractional Exponents?

When using fractional exponents, the power is the numerator and the root is the denominator. The typical rule for fractional exponents is as stated. x^m/n can be expressed as

3. What To Do With Negative Fractional Exponents?

If the exponent is specified as negative, we must remove the negative sign from the power and obtain the base’s reciprocal. 8^-1/2, for instance, equals (1/8)^1/2.

4. How To Solve Fractional Exponents?

We can easily multiply or divide numbers with fractional exponents by following to a few instructions.

• a^1/m × a^1/n = a^{(1/m + 1/n)}
• a^1/m ÷ a^1/n = a^{(1/m – 1/n)}
• a^1/m × b^1/m = (ab)^1/m
• a^1/m ÷ b^1/m = (a ÷ b)^1/m
• a^-m/n = (1/a)^m/n

5. How To Add Fractional Exponents?

There is no rule defined for the addition of fractional exponents. We can add them only by simplifying the powers, if possible. 

6. How To Divide Fractional Exponents?

When dividing fractional exponents with the same base and various powers, the powers are subtracted; however, when dividing exponents with different bases and the same powers, the bases are divided first, and the common power is then written on the solution.

7. How do We Simplify Expressions with Fractional Exponents?

To simplify, use rules like multiplying exponents when bases are the same, or convert fractional exponents to radicals and apply standard operations.