- The expression \((\sqrt{9})^2\) evaluates to \(9\) because \(\sqrt{9}=3\) and \(3^2=9\).
- The expression \(9^{1/2}\) is equivalent to \(\sqrt{9}\) based on the rule \(x^{1/2} = \sqrt{x}\).
- Final answers: \(\boxed{9}\) and \(\boxed{\sqrt{9}}\).
Explanation
- Simplifying the expression \((\sqrt{9})^2\)
We begin by simplifying the expression \((\sqrt{9})^2\). By the definition of the square root, \(\sqrt{x}\) is the non-negative number such that its square equals \(x\). Therefore, the square root of \(9\) is \(3\), because \(3 \times 3 = 9\). When we square this value, we have \(3^2 = 3 \times 3 = 9\). Thus, \((\sqrt{9})^2 = 9\).
- Identifying the radical form of \(9^{1/2}\)
Next, we consider the expression \(9^{1/2}\). The exponent rule for fractional powers states that \(x^{1/2} = \sqrt{x}\) for any non-negative \(x\). This relationship allows us to rewrite the fractional exponent as a radical. Applying this rule to \(9^{1/2}\), we obtain \(\sqrt{9}\). The question asks for the expression in radical form, and thus the equivalent form using the radical symbol is \(\sqrt{9}\).
- Conclusion
Combining our findings, we have determined that \((\sqrt{9})^2 = 9\) and \(9^{1/2} = \sqrt{9}\). These results fill the requested blanks appropriately.
Examples
In mathematics, radical form is often used to simplify or express numbers when dealing with geometric properties, such as finding the hypotenuse of a right-angled triangle using the Pythagorean theorem, where the length might be represented as \(\sqrt{a^2+b^2}\). Understanding the relationship between roots and fractional exponents is essential for working with more complex algebraic equations, as it allows us to convert between different types of representations efficiently. For example, in engineering, calculating the magnitude of a vector often requires converting square roots to their fractional power equivalents for computational efficiency.