Simplify the expression:
$\sqrt{45 y}+\sqrt{20 y}$
Assume that the variable represents a positive real number.
We are tasked with simplifying the expression \(\sqrt{45y} + \sqrt{20y}\) for a positive real number \(y > 0\). To simplify radical expressions of this form, we aim to extract the largest perfect square factor from each radicand. This allows us to combine terms that share the same radical component.
First, we factor each radicand to identify perfect square factors.
For \(\sqrt{45y}\), we observe that \(45 = 9 \times 5\). Thus,
\[\sqrt{45y} = \sqrt{9 \times 5y}.\]
For \(\sqrt{20y}\), we observe that \(20 = 4 \times 5\). Thus,
\[\sqrt{20y} = \sqrt{4 \times 5y}.\]
We apply the property of square roots \(\sqrt{ab} = \sqrt{a}\sqrt{b}\) for \(a, b \ge 0\).
Applying this to our expressions, we get:
\[\sqrt{45y} = \sqrt{9} \times \sqrt{5y} = 3\sqrt{5y}\]
\[\sqrt{20y} = \sqrt{4} \times \sqrt{5y} = 2\sqrt{5y}\]
Now, substitute these back into the original sum:
\[\sqrt{45y} + \sqrt{20y} = 3\sqrt{5y} + 2\sqrt{5y}.\]
Since both terms now share the same radical part \(\sqrt{5y}\), we can combine them by adding their coefficients:
\[(3 + 2)\sqrt{5y} = 5\sqrt{5y}.\]
Therefore, the simplified form of the expression is \(\boxed{5\sqrt{5y}}\).
Simplifying radical expressions is a fundamental skill in algebra that appears frequently in geometry and physics. For example, if you are calculating the distance between two points in a coordinate plane where the coordinates involve variables, or when determining the result of physical formulas like the period of a pendulum \(T = 2\pi\sqrt{\frac{L}{g}}\), being able to simplify square root expressions allows for cleaner, more manageable final results. Mastering these manipulations helps in standardizing expressions for further calculations.