Simplify the radical: $\sqrt{ 1 8 0 }$
To simplify the radical \(\sqrt{180}\), we aim to express the number under the square root, 180, as a product of a perfect square and another factor. This allows us to use the property \(\sqrt{x \cdot y} = \sqrt{x} \cdot \sqrt{y}\). First, we perform prime factorization of 180:
\[180 = 18 \times 10 = (2 \times 9) \times (2 \times 5) = 2 \times 3^2 \times 2 \times 5 = 2^2 \times 3^2 \times 5 = 4 \times 9 \times 5 = 36 \times 5.\]
We identify 36 as the largest perfect square factor.
Using the property \(\sqrt{x \cdot y} = \sqrt{x} \cdot \sqrt{y}\), we can rewrite the expression:
\[\sqrt{180} = \sqrt{36 \times 5} = \sqrt{36} \cdot \sqrt{5}.\]
Since the square root of 36 is 6, we have:
\[\sqrt{36} \cdot \sqrt{5} = 6\sqrt{5}.\]
Thus, the simplified form of \(\sqrt{180}\) is \(6\sqrt{5}\).
Now we compare our result with the given choices:
We have already found that \(\sqrt{180} = 6\sqrt{5}\), which matches option (A). We can verify the others:
None of these are equal to \(\sqrt{180}\). Therefore, the correct choice is (A).
Simplifying radicals is essential in many fields, including architecture and construction. For instance, if you need to calculate the length of a diagonal in a rectangular garden measuring \(\sqrt{180}\) units, you can use the simplified form to make measurements much more manageable. Instead of working with an unwieldy number, you can express it as \(6\sqrt{5}\), which helps you estimate or calculate precise dimensions more easily in your projects.