Simplify.
$\sqrt{75}$
$\sqrt{75}$
Simplify.
$\sqrt{75}$
We aim to simplify the radical expression \(\sqrt{75}\). Simplifying a square root involves identifying any perfect square factors within the radicand (the number under the radical symbol). A number is a perfect square if it is the result of multiplying an integer by itself (e.g., \(1, 4, 9, 16, 25, ...\)). Once the largest perfect square factor is identified, we can use the property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) to simplify the expression.
First, find the prime factorization of 75. Dividing 75 by the smallest prime, 3, we get:
\[75 = 3 \times 25\]
Since 25 is a perfect square (\(5^2 = 25\)), we can rewrite the radicand as the product of this perfect square and a remaining factor:
\[\sqrt{75} = \sqrt{25 \times 3}\]
Using the product property of radicals, \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), we can separate the terms:
\[\sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3}\]
Since the square root of 25 is 5, the expression simplifies to:
\[5 \times \sqrt{3} = 5\sqrt{3}\]
Thus, the simplified form of \(\sqrt{75}\) is \(5\sqrt{3}\).
Imagine you are a carpenter building a frame for a project. You need to calculate the length of a diagonal brace for a rectangle with sides of length 5 cm and 5 cm. By the Pythagorean theorem, the diagonal is \(\sqrt{5^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50}\). Simplifying radicals like these is essential for finding exact measurements in construction, carpentry, and design, ensuring that your cuts are accurate and your structures fit perfectly together. Another example is calculating the side length of a square area in geometry; if you know the area is 75 square units, the side length is \(\sqrt{75} = 5\sqrt{3}\) units.