What is the following sum?
$4 \sqrt{5}+2 \sqrt{5}$
The expression given is \(4 \sqrt{5} + 2 \sqrt{5}\). This expression involves the addition of two terms that share a common radical factor, \(\sqrt{5}\). In algebra, we can use the distributive property of real numbers, which states that \(a\sqrt{n} + b\sqrt{n} = (a+b)\sqrt{n}\). This allows us to combine like radical terms.
To simplify the expression \(4 \sqrt{5} + 2 \sqrt{5}\), we identify the coefficient of each term. The first term has a coefficient of \(4\) and the second term has a coefficient of \(2\). Since the radical part \(\sqrt{5}\) is the same for both, we can factor it out:
\[4 \sqrt{5} + 2 \sqrt{5} = (4 + 2) \sqrt{5}\]
Now, we perform the addition within the parentheses: \(4 + 2 = 6\).
Thus, the expression simplifies to \(6 \sqrt{5}\).
Comparing our result of \(6 \sqrt{5}\) to the provided options:
Our calculated result matches the third option. The correct simplification is \(\boxed{6 \sqrt{5}}\).
In real-life scenarios, adding radical expressions is common in geometry when calculating the perimeter of shapes involving square roots. For instance, if you are calculating the total length of a path made of two segments of \(\sqrt{5}\) meters and \(\sqrt{5}\) meters, you use this property to find the total: \(\sqrt{5} + \sqrt{5} = 2\sqrt{5}\) meters. Understanding how to combine these terms simplifies calculations for engineers, architects, and designers working with precise spatial dimensions.