- The expression \((\sqrt{9})^2\) simplifies to \(3^2 = 9\).
- Using the Power of a Power rule, \((9^{\frac{1}{2}})^2 = 9^1 = 9\).
- Since \(9^{\frac{1}{2}}\) is the square root of 9, it equals \(3\).
- Final answers: \((\sqrt{9})^2 = \boxed{9}\) and \(9^{\frac{1}{2}} = \boxed{3}\).
Explanation
- Evaluating the First Expression
We are given two expressions involving the square root and exponent rules. First, we examine \((\sqrt{9})^2\). The term \(\sqrt{9}\) represents the positive square root of \(9\), which is a number that, when multiplied by itself, results in \(9\). Since \(3 \times 3 = 9\), we have \(\sqrt{9} = 3\). Thus, squaring this result gives us \((\sqrt{9})^2 = 3^2 = 9\). The simplest form of \((\sqrt{9})^2\) is \(9\).
- Applying Power Rules and Concluding
Next, we consider the Power of a Power rule, which states that \((a^m)^n = a^{m \times n}\). We are given \(\left(9^{\frac{1}{2}}\right)^2 = 9^{\frac{1}{2} \times 2} = 9^{\frac{2}{2}} = 9^1 = 9\). Since squaring \(9^{\frac{1}{2}}\) yields \(9\), this confirms that \(9^{\frac{1}{2}}\) is equal to the square root of \(9\). Thus, \(9^{\frac{1}{2}} = \sqrt{9} = 3\). Therefore, the value of \(9^{\frac{1}{2}}\) is 3.
Examples
In real-world applications, understanding exponents and radicals is crucial for fields like architecture and engineering. For instance, if you need to calculate the side length of a square area (like a floor) of 9 square meters, you would use the square root: \(\sqrt{9} = 3\) meters. This demonstrates that radical expressions represent physical dimensions and are fundamental for spatial calculations.