Simplify.
$\sqrt{220}$
$\sqrt{220}$
Simplify.
$\sqrt{220}$
To simplify the radical \(\sqrt{220}\), we need to find the largest perfect square factor of the number 220. A perfect square factor is a factor of the form \(k^2\), where \(k\) is an integer. Once the largest perfect square factor is identified, we can use the property of radicals \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\) to simplify the expression.
First, let's perform the prime factorization of 220.
We can divide by 2:
\[220 = 2 \times 110\]
\[110 = 2 \times 55\]
Now, factor 55:
\[55 = 5 \times 11\]
Putting it all together, the prime factorization of 220 is:
\[220 = 2 \times 2 \times 5 \times 11 = 2^2 \times 55 = 4 \times 55\]
The largest perfect square factor of 220 is 4.
Now that we have identified the largest perfect square factor, we can rewrite the radical expression:
\[\sqrt{220} = \sqrt{4 \times 55}\]
Applying the property of radicals \(\sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b}\), we get:
\[\sqrt{4 \times 55} = \sqrt{4} \times \sqrt{55}\]
Since \(\sqrt{4} = 2\), the expression simplifies to:
\[2\sqrt{55}\]
Since 55 is the product of two distinct prime numbers (5 and 11), it has no square factors other than 1, so the expression is in its simplest form.
Simplifying square roots is a fundamental skill in geometry and algebra. For instance, if you are designing a rectangular garden with an area of 220 square feet, and you need to determine the side length based on a square root calculation for specific proportions, simplifying the root to \(2\sqrt{55}\) allows for a more precise estimation (\(\sqrt{55} \approx 7.416\), so the side is about \(14.832\) feet). This helps in accurately measuring materials or layout dimensions in real-world construction and landscaping projects.