To multiply radicals using the basic method, they have to have the same index. The "index" is the very small number written just to the left of the uppermost line in the radical symbol. If there is no index number, the radical is understood to be a square root (index 2) and can be multiplied with other square roots. You can multiply radicals with different indexes, but that is a more advanced method and will be explained later. Here are two examples of multiplication using radicals with the same indexes:   Ex. 1: √(18) x √(2) = ?  Ex. 2: √(10) x √(5) = ?  Ex. 3: 3√(3) x 3√(9) = ? Next, simply multiply the numbers under the radical or square root signs and keep them there. Here's how you do it:   Ex. 1: √(18) x √(2) = √(36)  Ex. 2: √(10) x √(5) = √(50)  Ex. 3: 3√(3) x 3√(9) = 3√(27) If you've multiplied radicals, there's a good chance that they can be simplified to perfect squares or perfect cubes, or that they can be simplified by finding a perfect square as a factor of the final product. Here's how you do it:   Ex. 1: √(36) = 6. 36 is a perfect square because it is the product of 6 x 6. The square root of 36 is simply 6.  Ex. 2: √(50) = √(25 x 2) = √([5 x 5] x 2) = 5√(2). Though 50 is not a perfect square, 25 is a factor of 50 (because it divides evenly into the number) and is a perfect square. You can break 25 down into its factors, 5 x 5, and move one 5 out of the square root sign to simplify the expression. You can think of it like this: If you throw the 5 back under the radical, it is multiplied by itself and becomes 25 again.   Ex. 3:3√(27) = 3. 27 is a perfect cube because it's the product of 3 x 3 x 3. The cube root of 27 is therefore 3.
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