Simplify the radical: $\sqrt{40}$
To simplify the expression \(\sqrt{40}\), we need to find the largest perfect square that is a factor of the radicand, which is 40. A perfect square is an integer that can be expressed as the product of an integer with itself (e.g., \(1, 4, 9, 16, 25, \dots\)). By identifying the largest perfect square factor, we can pull it out of the square root radical.
Let us list the positive integer factors of 40: 1, 2, 4, 5, 8, 10, 20, and 40. Among these factors, we look for the largest perfect square.
Therefore, the largest perfect square factor of 40 is 4. We can rewrite 40 as the product of this factor and its corresponding partner:
\[40 = 4 \times 10\]
Using the property of radicals \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\) for non-negative \(a\) and \(b\), we can simplify our expression:
\[\sqrt{40} = \sqrt{4 \times 10} = \sqrt{4} \times \sqrt{10}\]
Since \(\sqrt{4} = 2\), the expression becomes:
\[2 \times \sqrt{10} = 2\sqrt{10}\]
Comparing our result \(2\sqrt{10}\) to the given options:
The simplified form \(2\sqrt{10}\) matches option (C). Thus, the correct choice is (C).
Simplifying radicals is extremely useful in fields like engineering and architecture. For example, if you are designing a support beam and calculating the diagonal length of a frame using the Pythagorean theorem, you might end up with a value like \(\sqrt{40}\) meters. Expressing this as \(2\sqrt{10}\) meters gives a cleaner, more practical form for further calculations or material estimation, as you know \(\sqrt{10}\) is approximately 3.16, making the length roughly 6.32 meters. This skill helps turn complex roots into more manageable numbers.