Fractional Exponents (original) (raw)

Last Updated : 23 Apr, 2026

Fractional exponents, also known as rational exponents (radicals), are used to represent powers and roots together in a single expression. In an exponential expression of the form aᵇ, where a is the base and b is the exponent, if b is a fraction (m/n), it is called a fractional exponent.

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In a fractional exponent, the numerator (m) represents the power, and the denominator (n) represents the root.

**Here are some common examples of fractional exponents:

Exponent Name of the exponent Indication
1/2 Square root a^{\frac{1}{2}} = \sqrt{a}
1/3 Cube root a^{\frac{1}{3}} = \sqrt[3]{a}
1/4 Fourth root a^{\frac{1}{4}} = \sqrt[4]{a}

Rules of Fractional Exponents

**Rule 1: When multiplying powers with the same base, add the exponents: a^{\frac{1}{m}} \times a^{\frac{1}{n}} = a^{\frac{1}{m} + \frac{1}{n}}
**Rule 2: When dividing powers with the same base, subtract the exponents: a^{\frac{1}{m}} \div a^{\frac{1}{n}} = a^{\frac{1}{m} - \frac{1}{n}}
**Rule 3: When multiplying different bases with the same exponent, multiply the bases: a^{\frac{1}{m}} \times b^{\frac{1}{m}} = (ab)^{\frac{1}{m}}
**Rule 4: When dividing different bases with the same exponent, divide the bases: a^{\frac{1}{m}} \div b^{\frac{1}{m}} = \left(\frac{a}{b}\right)^{\frac{1}{m}}
**Rule 5: A negative exponent means taking the reciprocal: a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}}

**Example: Simplify (64/125)²⁄³

**Solution:

64 = 4³ and 125 = 5³

So, (64/125)²⁄³ = (4³/5³)²⁄³

= ((4/5)³)²⁄³ _(using law of exponents: (aᵐ)ⁿ = aᵐⁿ)

= (4/5)² _(since 3 × 2/3 = 2)

= 16/25

Fractional Exponents vs Integer Exponents

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

Fractional Exponents Integer Exponents
Used when the power is not an integer. Used when the powers 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

**Example: Solve 161/4

**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=161/4=2.

**Example: Solve 22/3 * 23/4

**Solution:

Multiply fractional exponents with same base

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

So, 22/3 x 23/4 = 217/12

**Example: Simplify 43/4 ÷ 45/8

**Solution:

In this case we express it as a1/m ÷ a1/n = a(1/m - 1/n).

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

**Example: Simplify 163/5 ÷ 43/5

**Solution:

Using a1/m ÷ b1/m = (a ÷ b)1/m

So, (16/4)(3/5) = 43/5

**Example: Simplify 49-1/2

**Solution:

The base is 49 and the exponent is −1/2.

First, remove the negative sign by taking the reciprocal of the base:
49⁻¹ᐟ² = (1/49)¹ᐟ²

Now, the exponent 1/2 means square root:
(1/49)¹ᐟ² = 1/√49

Since √49 = 7, we get:
1/√49 = 1/7

Solved Examples

**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 here 3 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 81/2 ÷ 21/2

**Solution:

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

Hence, we can solve this problem as, 81/2 ÷ 21/2 = (8/2)1/2 = 41/2 = 2.

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

31/2 × 31/4 × 31/8

**Solution:

The given expression can be re-written as,

31/2 × 31/4 × 31/8

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)

⇒ 37/8

Practice Problems

**Problem 1: Simplify:

**Problem 2: Evaluate:

**Problem 3: Calculate: