In the expression in which a is under a square root symbol and n is to the upper left of the square root symbol, we refer to the letter n as the index of the radical, and to a as the radicand. Radical expressions can only be multiplied or divided by one another if they have the same index. If they lack the same index (keeping in mind that this is the letter n, to the upper left of the radical symbol) then they cannot be multiplied or divided by one another. When we multiply radical expressions with like indices, only the radicand itself is affected. The index remains the same. So if we multiply 4 and 6, both of which are under radical signs, and the radicand of both is 5, the index remains the same in the product, although the radicand becomes 24.
The rule for adding and subtracting radical expressions is a little different. While we can multiply and divide radical expressions by one another providing the indices for both terms are identical, this is not true of adding and subtracting radical expresions. We can only add and subtract radical expressions if they have both the same index and radicand. When adding and subtracting radical expressions, both the radicands and the indices all remain the same. It is only the number to the left of the radical sign, that is outside of the radical sign but still part of the radical expression, that is altered. Indices and radicands always remain the same when added to one another or subtracted by one another.
Keep in mind that if we have a sum under one radical sign, this is not the same as when we have the same numbers in a sum whose components are under two distinct radical signs. 4 + 16 yields a different sum if both numbers are under distinct radical signs then if they are under the same radical sign. If they are under the same radical sign, then you have an answer that is the square root of 19. If they are under distinct radical signs, then you first find the square root of each component of the sum, and then add each result together. This means that you would determine that the square root of 4 is 2 and the square root of 16 is 4, and then you would add 2 and 4 together, and you would get 6.
We will oftentimes need to simplify radical expressions. This is helpful where we have a number in which the product taken to produce the number has as one of its components a number with a perfect square. Keep in mind that a perfect square is a number that can be produced when a number is multiplied by itself. For example, 9 is a perfect square because it can be obtained by multiplying 3 by itself. What we try to do when simplifying radical expresions is breaking a number up into a product in which one of the components of the product is a perfect square. Take the case of the square root of 63. 9 x 7 = 63. 9 is a perfect square, but 7 is not. We would put each component of the product under a square root symbol. The square root of 9 would reduce to 3 but 7 would remain under the square root because it cannot be reduced. It cannot be reduced because it is not a perfect square. Therefore, we would write the simplified radical expression as 3 times the square root of 7 (7 would remain under the square root symbol).