What is $\sqrt{50}$ in simplest radical form?
We are tasked with simplifying the expression \(\sqrt{50}\). To write a square root in simplest radical form, we must extract all perfect square factors from the radicand (the number under the radical). A radical is in simplest form when the radicand has no perfect square factors other than 1.
The first step is to find the prime factorization of 50. We know that 50 is an even number, so it is divisible by 2:
\[50 = 2 \times 25\]
Since 25 is a perfect square (\(5^2\)), we can rewrite the expression as:
\[\sqrt{50} = \sqrt{25 \times 2}\]
Using the property of radicals \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), we can separate the terms:
\[\sqrt{50} = \sqrt{25} \times \sqrt{2}\]
Since the square root of 25 is 5, we have:
\[\sqrt{50} = 5\sqrt{2}\]
The number 2 has no perfect square factors other than 1, so the radical is now in its simplest form.
The simplified form of \(\sqrt{50}\) is \(5\sqrt{2}\). This value can be verified by squaring it: \((5\sqrt{2})^2 = 5^2 \times (\sqrt{2})^2 = 25 \times 2 = 50\). Thus, our simplification is correct. The final answer is \(\boxed{5\sqrt{2}}\).
Simplifying radicals is essential in geometry and physics. For example, if you are calculating the length of the diagonal of a square with a side length of 5, the Pythagorean theorem tells you the diagonal length is \(\sqrt{5^2 + 5^2} = \sqrt{50}\). Knowing that \(\sqrt{50} = 5\sqrt{2}\) allows you to represent the distance more precisely and perform further calculations, such as determining the perimeter of larger shapes or analyzing waveforms in engineering, more efficiently.