What is the following product?
$\sqrt{10} \cdot \sqrt{10}$
The problem asks for the product of two square roots, specifically \(\sqrt{10} \cdot \sqrt{10}\). We start by recalling the definition of a square root. For any non-negative real number \(x\), the square root \(\sqrt{x}\) is defined as the unique non-negative number such that when multiplied by itself, the result is \(x\). Symbolically, this means \((\sqrt{x})^2 = \sqrt{x} \cdot \sqrt{x} = x\). In our case, \(x = 10\), which is a non-negative real number. Thus, applying this fundamental definition directly gives \(\sqrt{10} \cdot \sqrt{10} = 10\).
Alternatively, we can use the property of radicals that allows us to multiply two square roots with the same radicand: \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\). Applying this property to our expression, we have \(\sqrt{10} \cdot \sqrt{10} = \sqrt{10 \cdot 10} = \sqrt{100}\). Since \(\sqrt{100} = 10\), we reach the same result. Both methods confirm that the product is 10. Comparing this with the provided choices, it corresponds to the first option.
Understanding how square roots behave is essential in geometry and physics. For example, if you need to calculate the area of a square-shaped plot of land where each side length is \(\sqrt{10}\) meters, the area is calculated as side times side, or \(\sqrt{10} \times \sqrt{10} = 10\) square meters. This principle is widely used whenever you deal with distances or diagonal lengths in various engineering and design applications.