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:

Practice Questions on Fractional Exponents