In this Section:
In this section, we will learn how to simplify radicals. We begin by learning about the product rule for radicals, along with the quotient rule for radicals. The product rule for radicals tells us if the index is the same, we can multiply the radicands and place the result under a radical with the same index. This process can also be reversed to help us simplify. If we saw an example such as the square root of 20, we could factor 20 and re-write this problem as: the square root of 4 multiplied by the square root of 5. Once we have done this, we can see that the square root of 4 represents a rational number. We can display this as 2, versus the square root of 4. The quotient rule for radicals is fairly similar. When we have division under the radical symbol, we can split this up into two separate radicals. As an example, the square root of (25/4) could be re-written as the square root of 25 divided by the square root of 4. We can then simplify this to 5 / 2 or 2.5 in decimal form. When we simplify radicals, we want to meet four general conditions: 1) the radicand contains no factor that is greater than or equal to the index. 2) The radicand has no fractions. 3) No denominator contains a radical. 4) There can be no common factor between the index of the radical and the exponent in the radicand.