Which expression is the exponential form of $\sqrt{10}$?
The expression we need to convert is \(\sqrt{10}\). This is a radical expression representing the square root of 10. By definition, a radical expression \(\sqrt[n]{x}\) represents the \(n\)-th root of \(x\). When the index \(n\) is not explicitly written in a square root symbol, it is understood to be 2. Therefore, \(\sqrt{10} = \sqrt[2]{10}\).
The property of exponents relating to radicals states that any radical \(\sqrt[n]{x}\) can be rewritten in exponential form as \(x^{\frac{1}{n}}\). In our case, where \(x = 10\) and \(n = 2\), applying this property yields \(\sqrt[2]{10} = 10^{\frac{1}{2}}\). This exponential form represents the square root of 10.
Comparing our derived result \(10^{\frac{1}{2}}\) with the given choices:
Thus, the correct expression is \(10^{\frac{1}{2}}\). The final answer is \(\boxed{10^{\frac{1}{2}}}\).
Understanding how to convert radicals to exponents is fundamental in fields like engineering and physics. For example, if you are working with formulas that calculate energy or distance, such as the relationship between speed and kinetic energy or the Pythagorean theorem, you often need to manipulate powers to simplify equations. Knowing that \(\sqrt{x}\) is equivalent to \(x^{1/2}\) allows you to use the rules of exponents to solve complex algebraic problems more efficiently, helping you calculate physical quantities with greater ease.