The principle that permits the division of radicals before simplifying is based on the property of square roots (or radicals in general) that allows us to separate the numerator and denominator. This property can be expressed as follows:
\[ \frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \quad \text{for } b \neq 0 \]
This means that if you have a fraction involving square roots, you can split the square root of the numerator and denominator. For example:
\[ \frac{\sqrt{8}}{\sqrt{2}} = \sqrt{\frac{8}{2}} = \sqrt{4} = 2 \]
This principle holds for all positive numbers under the radical and allows one to simplify expressions involving radicals more easily. However, it's important to ensure the quantities involved are non-negative when applying this property, as square roots of negative numbers are not defined in the set of real numbers.
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