How to multiply fractions with exponents and variables

Learning to deal with exponents forms an integral part of any math education, but thankfully the rules for multiplying and dividing them match the rules for non-fractional exponents. The first step to understanding how to deal with fractional exponents is getting a rundown of what exactly they are, and then you can look at the ways you can combine exponents when they’re multiplied or divided and they have the same base. In brief, you add the exponents together when multiplying and subtract one from the other when dividing, provided they have the same base.

TL;DR (Too Long; Didn't Read)

Multiply terms with exponents using the general rule:

​xa​ + ​xb​ = ​x​(​a​ + ​b​)

And divide terms with exponents using the rule:

​xa ​÷ ​xb​ = ​x​(​a​ – ​b​)

These rules work with any expression in place of ​a​ and ​b​, even fractions.

What Are Fractional Exponents?

Fractional exponents provide a compact and useful way of expressing square, cube and higher roots. The denominator on the exponent tells you what root of the “base” number the term represents. In a term like ​xa​, you call ​x​ the base and ​a​ the exponent. So a fractional exponent tells you:

x^{1/2} = \sqrt{x}

The denominator of two on the exponent tells you that you’re taking the square root of ​x​ in this expression. The same basic rule applies to higher roots:

x^{1/3} = \sqrt[3]{x}

And

x^{1/4} = \sqrt[4]{x}

This pattern continues. For a concrete example:

9^{1/2} = \sqrt{9}=3

And

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

Fraction Exponent Rules: Multiplying Fractional Exponents With the Same Base

Multiply terms with fractional exponents (provided they have the same base) by adding together the exponents. For example:

x^{1/3} × x^{1/3} × x^{1/3} = x^{(1/3 + 1/3 + 1/3)} \\ = x^1 = x

Since ​x​1/3 means “the cube root of ​x​,” it makes perfect sense that this multiplied by itself twice gives the result ​x​. You may also run into examples like ​x​1/3 × ​x​1/3, but you deal with these in exactly the same way:

x^{1/3} × x^{1/3} = x^{( 1/3 + 1/3)} \\ = x^{2/3}

The fact that the expression at the end is still a fractional exponent doesn’t make a difference to the process. This can be simplified if you note that ​x​2/3 = (​x​1/3)2 = ∛​x​2. With an expression like this, it doesn’t matter whether you take the root or the power first. This example illustrates how to calculate these:

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

Since the cube root of 8 is easy to work out, tackle this as follows:

(\sqrt[3]{8})^2 = 2^2 = 4

So this means:

8^{1/3} + 8^{1/3}= 4

You may also encounter products of fractional exponents with different numbers in the denominators of the fractions, and you can add these exponents in the same way you’d add other fractions. For example:

\begin{aligned} x^{1/4} × x^{1/2} &= x^{(1/4 + 1/2)} \\ &= x^{(1/4 + 2/4)} \\ &= x^{3/4} \end{aligned}

These are all specific expressions of the general rule for multiplying two expressions with exponents:

x^a + x^b = x^{(a + b)}

Fraction Exponent Rules: Dividing Fractional Exponents With the Same Base

Tackle divisions of two numbers with fractional exponents by subtracting the exponent you’re dividing (the divisor) by the one you’re dividing (the dividend). For example:

x^{1/2} ÷ x^{1/2} = x^{(1/2 - 1/2)} \\ = x^0 = 1

This makes sense, because any number divided by itself equals one, and this agrees with the standard result that any number raised to a power of 0 equals one. The next example uses numbers as bases and different exponents:

\begin{aligned} 16^{1/2} ÷ 16^{1/4} &= 16^{(1/2 - 1/4)} \\ &= 16^{(2/4 - 1/4)} \\ &= 16^{1/4} \\ &= 2 \end{aligned}

Which you can also see if you note that 161/2 = 4 and 161/4 = 2.

As with multiplication, you may also end up with fractional exponents that have a number other than one in the numerator, but you deal with these in the same way.

These simply express the general rule for dividing exponents:

x^a ÷ x^b = x^{(a - b)}

Multiplying and Dividing Fractional Exponents in Different Bases

If the bases on the terms are different, there is no easy way to multiply or divide exponents. In these cases, simply calculate the value of the individual terms and then perform the required operation. The only exception is if the exponent is the same, in which case you can multiply or divide them as follows:

x^4 × y^4 = (xy)^4 \\ x^4 ÷ y^4 = (x ÷ y)^4

If an exponent of a number is a fraction, it is called a fractional exponent. Exponents show the number of times a number is replicated in multiplication. For example, 42 = 4×4 = 16. Here, exponent 2 is a whole number. In the number, say x1/y, x is the base and 1/y is the fractional exponent.

In this article, we will discuss the concept of fractional exponents, and their rules, and learn how to solve them. We shall also explore negative fractional exponents and solve various examples for a better understanding of the concept. 

What are Fractional Exponents?

Fractional exponents are ways to represent powers and roots together. 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. A few examples of fractional exponents are 21/2, 32/3, etc. The general form of a fractional exponent is xm/n, where x is the base and m/n is the exponent.

Look at the figure given below to understand how fractional exponents are represented.

Some examples of fractional exponents that are widely used are given below:

Fractional Exponents Rules

There are certain rules to be followed that help us to multiply or divide numbers with fractional exponents easily. Many people are familiar with whole-number exponents, but when it comes to fractional exponents, they end up doing mistakes that can be avoided if we follow these rules of fractional exponents.

• Rule 1: a1/m × a1/n = a(1/m + 1/n)
• Rule 2: a1/m ÷ a1/n = a(1/m - 1/n)
• Rule 3: a1/m × b1/m = (ab)1/m
• Rule 4: a1/m ÷ b1/m = (a÷b)1/m
• Rule 5: a-m/n = (1/a)m/n

These rules are very helpful while simplifying fractional exponents. Let us now learn how to simplify fractional exponents.

Simplifying Fractional Exponents

Simplifying fractional exponents can be understood in two ways which are multiplication and division. It involves reducing the expression or the exponent to a reduced form that is easy to understand. For example, 91/2 can be reduced to 3. Let us understand the simplification of fractional exponents with the help of some examples.

1) Solve 3√8 = 81/3

We know that 8 can be expressed as a cube of 2 which is given as, 8 = 23. Substituting the value of 8 in the given example we get, (23)1/3 = 2 since the product of the exponents gives 3×1/3=1. ∴ 3√8=81/3=2.

2) Simplify (64/125)2/3

In this example, both the base and the exponent are in fractional form. 64 can be expressed as a cube of 4 and 125 can be expressed as a cube of 5. They are given as, 64=43 and 125=53. Substituting their values in the given example we get, (43/53)2/3. 3 is a common power for both the numbers, hence (43/53)2/3 can be written as ((4/5)3)2/3, which is equal to (4/5)2 as 3×2/3=2. Now, we have (4/5)2, which is equal to 16/25. Therefore, (64/125)2/3 = 16/25.

Multiply Fractional Exponents With the Same Base

To multiply fractional exponents with the same base, we have to add the exponents and write the sum on the common base. The general rule for multiplying exponents with the same base is a1/m × a1/n = a(1/m + 1/n). For example, to multiply 22/3 and 23/4, we have to add the exponents first. So, 2/3 + 3/4 = 17/12. Therefore, 22/3 × 23/4 = 217/12.

How to Divide Fractional Exponents?

The division of fractional exponents can be classified into two types.

• Division of fractional exponents with different powers but the same bases
• Division of fractional exponents with the same powers but different bases

When we divide fractional exponents with different powers but the same bases, we express it as a1/m ÷ a1/n = a(1/m - 1/n). Here, we have to subtract the powers and write the difference on the common base. For example, 53/4 ÷ 51/2 = 5(3/4-1/2), which is equal to 51/4.

When we divide fractional exponents with the same powers but different bases, we express it as a1/m ÷ b1/m = (a÷b)1/m. Here, we are dividing the bases in the given sequence and writing the common power on it. For example, 95/6 ÷ 35/6 = (9/3)5/6, which is equal to 35/6.

Negative Fractional Exponents

Negative fractional exponents are the same as rational exponents. In this case, along with a fractional exponent, there is a negative sign attached to the power. For example, 2-1/2. To solve negative exponents, we have to apply exponents rules that say a-m = 1/am. It means before simplifying an expression further, the first step is to take the reciprocal of the base to the given power without the negative sign. The general rule for negative fractional exponents is a-m/n = (1/a)m/n.

For example, let us simplify 343-1/3. Here the base is 343 and the power is -1/3. The first step is to take the reciprocal of the base, which is 1/343, and remove the negative sign from the power. Now, we have (1/343)1/3. As we know that 343 is the third power of 7 as 73 = 343, we can re-write the expression as 1/(73)1/3. Since 3 and 1/3 cancel each other, the final answer is 1/7.

Related Articles

• Exponents
• Non-Integer Rational Exponents
• Irrational Exponents
• Exponential Terms
• Negative Exponents

FAQs on Fractional Exponents

What Do Fractional Exponents Mean?

Fractional exponents mean the power of a number is in terms of fraction rather than an integer. For example, in am/n the base is 'a' and the power is m/n which is a fraction.

What is the Rule for Fractional Exponents?

In the case of fractional exponents, the numerator is the power and the denominator is the root. This is the general rule of fractional exponents. We can write xm/n as n√(xm).

What To Do With Negative Fractional Exponents?

If the exponent is given in negative, it means we have to take the reciprocal of the base and remove the negative sign from the power. For example, 2-1/2 = (1/2)1/2.

How To Solve Fractional Exponents?

To solve fractional exponents, we use the laws of exponents or the exponent rules. The fractional exponents' rules are stated below:

• Rule 1: a1/m × a1/n = a(1/m + 1/n)
• Rule 2: a1/m ÷ a1/n = a(1/m - 1/n)
• Rule 3: a1/m × b1/m = (ab)1/m
• Rule 4: a1/m ÷ b1/m = (a÷b)1/m
• Rule 5: a-m/n = (1/a)m/n

How To Add Fractional Exponents?

There is no rule for the addition of fractional exponents. We can add them only by simplifying the powers, if possible. For example, 91/2 + 1251/3 = 3 + 5 = 8.

How To Divide Fractional Exponents?

Division of fractional exponents with the same base and different powers is done by subtracting the powers, and the division with different bases and same powers is done by dividing the bases first and writing the common power on the answer.

How do you cross multiply variables with exponents?