Rational Exponents
We can also use exponential notation to represent a radical. To do so, we
use a rational exponent.
Recall that a rational number is a number that may be written as the ratio
of two integers.
So, a rational exponent may be a fraction, such as
, or an integer
such as
.
Definition â€” Rational Exponent With Numerator 1
If a is a real number, then
Here, n is a positive integer.
If n is odd, then a^{1/n} is a real number.
If n is even, then a^{1/n} is a real number only when a
≥ 0.
Using this definition, we write:
Note:
The way we interpret 2n depends upon
the value of n:
â€¢ n is a whole number
â€¢ n is zero
2^{0} = 1
â€¢ n is a negative integer
â€¢ n is a fraction
Example
Rewrite using a radical and evaluate: 125^{1/3}
Solution
Write the radical symbol.


The index is 3, the denominator of the exponent


The radicand is 125, the base of the exponential expression.


Find the prime factorization of 125.
125 = 5 Â· 5
Â· 5 = 5^{3}.
Since 5^{3} = 125, the cube root of 125 is 5.
So,
