The expression $9^{(1/2)}$ equals $3$.

To understand why $9^{(1/2)}$ is equal to $3$, we need to explore the concepts of exponents and roots. The expression $9^{(1/2)}$ can be interpreted as the square root of $9$. In mathematical terms, the square root of a number is defined as the value that, when multiplied by itself, yields the original number. For instance, the square root of $9$ is a number that satisfies the equation $x^2 = 9$.

When we write $9^{(1/2)}$, the exponent $\frac{1}{2}$ indicates the operation of taking the square root. This is because raising a number to the power of $\frac{1}{2}$ is equivalent to finding its square root. Thus, we can rewrite $9^{(1/2)}$ as $\sqrt{9}$.

Next, let’s determine the square root of $9$. We need to identify which number, when multiplied by itself, equals $9$. The number $3$ meets this criterion, as $3 \times 3 = 9$. Therefore, we conclude that the square root of $9$ is $3$, confirming that $9^{(1/2)}$ is indeed equal to $3$.

Grasping this concept is essential, as it lays the groundwork for understanding more advanced operations involving exponents and roots. Keep in mind that the notation and rules governing exponents are consistent, which means that once you master this concept, you can apply it to other numbers and various types of roots, such as cube roots (which correspond to an exponent of $\frac{1}{3}$) and beyond.