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.
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.
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.
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 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.