When studying the exponential growth of a certain organism (such as blue-green algae), if the growth rate is $6.25\%$, the population after $x$ days can be expressed as $y = (1+6.25\%)^x$. But what if $x$ is not an integer (e.g., $1.5$ days)? Does this formula still make sense? To answer this, we must extend the definition of exponents from integers to rational numbers and even real numbers—this is an inevitable requirement of number system expansion.
$n$-th Roots and Fractional Exponents
Definition of $n$-th Root: Generally, if $x^n = a$, then $x$ is called the $n$-th root of $a$, where $n > 1$ and $n \in \mathbf{N}^*$. The expression $\sqrt[n]{a}$ is called a radical.
Fractional Exponent Powers: To unify operational properties, we define the positive fractional exponent of a positive number as: $\sqrt[n]{a^m} = a^{\frac{m}{n}}$ for $a > 0$. This means all radicals can be converted into power forms for computation.
Radicals are the manifestation of power operations in fractional dimensions. By defining fractional exponents, we eliminate the boundary between radicals and exponents, allowing operations to be unified.
$$(\sqrt[n]{a})^n = a, \quad \sqrt{b} = b^{\frac{1}{2}} \text{ for } b > 0$$