When dividing square roots, it is often beneficial to divide the radicands before simplifying the fraction when the radicands are both positive and simplify neatly. Specifically, if you have a fraction of the form \(\frac{\sqrt{a}}{\sqrt{b}}\), you can apply the property of square roots that states:
\[ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \]
This rule allows you to combine the square roots into a single square root of the fraction formed by dividing the radicands. This can sometimes simplify the expression more easily, especially if \(a\) and \(b\) have common factors, or if the fraction \(\frac{a}{b}\) can be expressed as a perfect square.
To use this approach effectively, consider the following:
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Simplifying the Radicands: If the numbers \(a\) and \(b\) can be simplified (e.g., if they share common factors or if \(b\) divides \(a\)), combining them under a single square root can make it easier to simplify further.
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Reducing Complexity: If \(b\) is a perfect square, or if \(\frac{a}{b}\) results in a simpler expression than working with \(\sqrt{a}\) and \(\sqrt{b}\) separately, it is beneficial to use the division of radicands.
In summary, the rule that allows the division of radicands is rooted in the basic properties of square roots, specifically \(\sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}}\), and is advantageous when it leads to simplification of the root or the expression itself.
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