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The product rule for radicals is a fundamental property used to simplify expressions involving radicals. This rule states that when two numbers under the same radical sign are multiplied, the result is the same as multiplying the two radicals together. Mathematically, it can be written as: \[\sqrt[m]{a} \times \sqrt[m]{b} = \sqrt[m]{a \times b}\] where \(m\) is the index of the radical and \(a\) and \(b\) are the radicands, which are the numbers under the radical signs.

Before we dive into the product rule, let's understand the terms involved: 1. Radicals: Radicals are mathematical expressions that show the root of a number. The most common radicals are square roots (\(\sqrt{}\) ), cube roots (\(\sqrt[3]{}\) ), and so on. 2. Radicand: The number under the radical sign is called the radicand. For example, in \(\sqrt{9}\), the radicand is 9. 3. Index: The index indicates the degree of the root. For example, in a square root (\(\sqrt{}\)), the index is 2, and in a cube root (\(\sqrt[3]{}\)), the index is 3. If the index is not specifically written, it is assumed to be 2 (a square root). Now that we have a basic understanding of the terms involved, we can explore the product rule for radicals.

Let's consider an example to demonstrate the application of the product rule: Example: Multiply \(\sqrt{2}\) by \(\sqrt{3}\). Using the product rule for radicals, we can multiply the two radicals as follows: \[\sqrt{2}\times\sqrt{3} = \sqrt{2 \times 3} = \sqrt{6}\] Therefore, the result of multiplying \(\sqrt{2}\) and \(\sqrt{3}\) is \(\sqrt{6}\).

The product rule is not limited to square roots; it can be applied to radicals with higher indexes as well. Let's consider another example: Example: Multiply \(\sqrt[3]{4}\) by \(\sqrt[3]{9}\). Using the product rule for radicals with the same index, we can multiply the two cube roots as follows: \[\sqrt[3]{4} \times \sqrt[3]{9} = \sqrt[3]{4 \times 9} = \sqrt[3]{36}\] Therefore, the result of multiplying \(\sqrt[3]{4}\) and \(\sqrt[3]{9}\) is \(\sqrt[3]{36}\).

In summary, the product rule for radicals states that when two numbers under the same radical sign are multiplied, the result is the same as multiplying the two radicals together. This rule applies to radicals of any index, and it is essential for simplifying expressions involving radicals. By understanding and applying the product rule, one can easily handle various mathematical problems involving radicals in a more efficient and simplified manner.