- Determine the prime factorization of 75 as \(3 \times 5^2\).
- Identify the largest perfect square factor, which is 25.
- Apply the radical property \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) to get \(\sqrt{25 \times 3} = 5\sqrt{3}\).
- The simplified form is \(\boxed{5\sqrt{3}}\).
Explanation
- Prime Factorization of 75
To simplify the square root of 75, we need to find the prime factorization of 75. A prime factorization decomposes a number into a product of prime numbers. We observe that 75 is divisible by 5: \(75 = 5 \times 15\). Further, 15 is divisible by 5: \(15 = 5 \times 3\). Thus, the prime factorization of 75 is \(3 \times 5 \times 5\), or \(3 \times 5^2\).
- Simplifying the Radical
To simplify the radical \(\sqrt{75}\), we look for the largest perfect square factor of 75. From the prime factorization \(3 \times 5^2\), we can see that \(5^2 = 25\) is a perfect square factor of 75. Therefore, we can rewrite the expression as \(\sqrt{25 \times 3}\). Using the product property of radicals, which states that \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) for non-negative \(a\) and \(b\), we have \(\sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3}\). Since \(\sqrt{25} = 5\), the expression simplifies to \(5\sqrt{3}\).
Examples
In geometry, if you have a right-angled triangle with two legs of length 5 and 5\sqrt{2}, the hypotenuse is \(\sqrt{5^2 + (5\sqrt{2})^2} = \sqrt{25 + 50} = \sqrt{75}\). Simplifying this to \(5\sqrt{3}\) makes it much easier to compare or add the length to other segments in drawings or engineering calculations, providing a more intuitive sense of its value.