Dividing the radicands before simplifying a fraction involving square roots can be advantageous in certain situations, particularly when it simplifies the expression more clearly or makes the simplification process easier.
The principle that permits this division is based on the properties of square roots, specifically the property that states:
\[ \sqrt{a} / \sqrt{b} = \sqrt{a / b} \quad \text{(for } b \neq 0\text{)} \]
This property indicates that you can simplify the quotient of two square roots by combining the radicands into a single square root. This is often beneficial in the following circumstances:
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Simplifying Complex Radicals: If the radicands contain products, sums, or terms that can be factored easily, dividing them can make it simpler to reduce the square root or manage complex expressions.
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Rationalizing Denominators: When you are working with a fraction where the denominator includes a square root, dividing the radicands can help to rewrite the expression so that it becomes simpler or allows for easier rationalization.
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Avoiding Complicated Radicals: Sometimes, writing the square root of a quotient can make it simpler to work with than manipulating the square root of complex sums or differences, particularly when variables or very large numbers are involved.
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One Radical is Easily Simplified: If you find that the radical in the numerator can be simplified significantly by utilizing the square root property before attempting to simplify the whole fraction, this can save time and effort.
Example
Consider the fraction:
\[ \frac{\sqrt{18}}{\sqrt{2}} \]
Instead of simplifying each radical separately, we can combine them:
\[ \frac{\sqrt{18}}{\sqrt{2}} = \sqrt{\frac{18}{2}} = \sqrt{9} = 3 \]
This way, we arrive at a much simpler solution than if we had tried to simplify \(\sqrt{18}\) and \(\sqrt{2}\) separately first.
Conclusion
In summary, it is advantageous to divide radicands when it leads directly to a simpler equivalent expression, permits easier manipulation, or when it makes rationalizing a denominator more straightforward. The foundational principle that allows this division is the property of square roots related to division.
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