Simplify.
$\sqrt{75 w}-\sqrt{12 w}$
Assume that the variable represents a positive real number.
The goal is to simplify the expression \(\sqrt{75w} - \sqrt{12w}\), where \(w\) is a positive real number. To simplify this, we need to express the radicands (the values inside the square roots) in a form that involves a perfect square factor. This allows us to extract the root of that factor, leading to terms that contain the same radical part, which can then be combined.
We start by factoring the numerical coefficients inside the radicals to identify any perfect square factors.
For 75, we note that \(75 = 25 \times 3\), where 25 is a perfect square (\(5^2\)).
For 12, we note that \(12 = 4 \times 3\), where 4 is a perfect square (\(2^2\)).
Applying these to the given radicals:
\[\sqrt{75w} = \sqrt{25 \times 3w} = \sqrt{25} \times \sqrt{3w} = 5\sqrt{3w}\]
\[\sqrt{12w} = \sqrt{4 \times 3w} = \sqrt{4} \times \sqrt{3w} = 2\sqrt{3w}\]
Now that we have rewritten both terms in the form of a coefficient multiplied by \(\sqrt{3w}\), the expression becomes:
\[5\sqrt{3w} - 2\sqrt{3w}\]
Since both terms are now "like radicals," we can subtract their coefficients:
\[(5 - 2)\sqrt{3w} = 3\sqrt{3w}\]
Thus, the expression simplifies to \(3\sqrt{3w}\).
Simplifying radicals is a fundamental skill used in various fields like architecture, engineering, and physics. For instance, if you are calculating the length of a diagonal in a room with a specific base area, you might encounter expressions involving square roots. By simplifying them as shown here, you can more easily estimate lengths, compare distances, and work with formulas in a more efficient and readable way.