Simplify.
$3 \sqrt{245}$
$3 \sqrt{245}$
Simplify.
$3 \sqrt{245}$
The goal is to simplify the expression \(3\sqrt{245}\). To simplify a square root, we look for factors of the radicand (the number inside the radical, 245) that are perfect squares. A perfect square is an integer that is the square of another integer, such as \(1, 4, 9, 16, 25, 36, 49, \dots\). By identifying the largest perfect square factor, we can pull it out of the radical.
To find the factors of 245, we can look at its prime factorization. Since 245 ends in 5, it is divisible by 5:
\[245 = 5 \times 49\]
We recognize that 49 is a perfect square because \(49 = 7^2\). Thus, we can rewrite the radicand as:
\[245 = 5 \times 7^2\]
Using the property of radicals \(\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}\), we can simplify the square root of 245:
\[\sqrt{245} = \sqrt{49 \times 5} = \sqrt{49} \times \sqrt{5} = 7\sqrt{5}\]
Now, substitute this result back into the original expression \(3\sqrt{245}\):
\[3\sqrt{245} = 3 \times (7\sqrt{5})\]
Multiplying the coefficients, we get:
\[3 \times 7 = 21\]
So, the expression simplifies to \(21\sqrt{5}\).
Simplifying square roots is a fundamental skill in algebra that allows for easier arithmetic and comparison of values. For example, if you are designing a square garden with an area of 245 square meters, you might need to calculate the side length. While \(\sqrt{245}\) is technically the side length, expressing it as \(7\sqrt{5}\) (which is approximately \(7 \times 2.236 \approx 15.65\) meters) makes it much easier to estimate and work with during construction planning. This method of simplifying radicals is used constantly in fields ranging from physics and engineering to architecture.