Fractional Exponents

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.

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:

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:

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.

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

Solution:

Multiply 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

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)

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

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

Related Articles

• Laws of Exponents
• Laws of Logarithms
• Comparing Fractions
• Addition of Fractions
• Subtracting Fractions

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 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

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