The principle that permits the division of radicals is based on the properties of exponents and the notion of simplifying radical expressions. Specifically, when dealing with radicals, you can use the following property:
\[ \sqrt{a} \div \sqrt{b} = \sqrt{\frac{a}{b}} \quad (\text{for } b \neq 0) \]
This means that the square root of a quotient is equal to the quotient of the square roots. This property holds true for all types of roots as well, not just square roots. Thus, you can generalize it as follows:
\[ \sqrt[n]{a} \div \sqrt[n]{b} = \sqrt[n]{\frac{a}{b}} \quad (\text{for } b \neq 0) \]
This principle allows us to simplify expressions involving radicals, making it easier to compute or manipulate them. It is important to note that this is valid under the assumption that the numbers involved are positive (or within the domain of the radical operation).
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