nth root

From this definition we can see that a radical is simply another notation for the first rational exponent that we looked at in the rational exponents section . Note as well that the index is required in these to make sure that we correctly evaluate the radical. There is one exception to this rule and that is square root. For square roots we have, From our discussion of exponents in the previous sections we know that only the term immediately to the left of the exponent actually gets the exponent. Therefore, the radical form of this is, So, we once again see that parenthesis are very important in this class. Be careful with them. So, the index is important. Different indexes will give different evaluations so make sure that you don’t drop the index unless it is a 2 (and hence we’re using square roots). This may not seem to be all that important, but in later topics this can be very important. Following this convention means that we will always get predictable values when evaluating roots. Note that we don’t have a similar rule for radicals with odd indexes such as the cube root in part (d) above. This is because there will never be more than one possible answer for a radical with an odd index. If you aren’t sure that you believe this consider the following quick number example. So, we’ve got the radicand written as a perfect square times a term whose exponent is smaller than the index. The radical then becomes, Now use the second property of radicals to break up the radical and then use the first property of radicals on the first term. This now satisfies the rules for simplification and so we are done. This radical violates the second simplification rule since both the index and the exponent have a common factor of 3. To fix this all we need to do is convert the radical to exponent form do some simplification and then convert back to radical form. Now that we’ve got a couple of basic problems out of the way let’s work some harder ones. Although, with that said, this one is really nothing more than an extension of the first example. There is more than one term here but everything works in exactly the same fashion. We will break the radicand up into perfect squares times terms whose exponents are less than 2 ( i.e. 1). Don’t forget to look for perfect squares in the number as well. Note that we used the fact that the second property can be expanded out to as many terms as we have in the product under the radical. Also, don’t get excited that there are no 𝑥’s under the radical in the final answer. This will happen on occasion. In this case don’t get excited about the fact that all the 𝑦’s stayed under the radical. That will happen on occasion. This last part seems a little tricky. Individually both of the radicals are in simplified form. However, there is often an unspoken rule for simplification. The unspoken rule is that we should have as few radicals in the problem as possible. In this case that means that we can use the second property of radicals to combine the two radicals into one radical and then we’ll see if there is any simplification that needs to be done. Before moving into a set of examples illustrating the last two simplification rules we need to talk briefly about adding/subtracting/multiplying radicals. Performing these operations with radicals is much the same as performing these operations with polynomials. If you don’t remember how to add/subtract/multiply polynomials we will give a quick reminder here and then give a more in depth set of examples the next section. With radicals we multiply in exactly the same manner. The main difference is that on occasion we’ll need to do some simplification after doing the multiplication Don’t get excited about the fact that there are two variables here. It works the same way! Again, notice that we combined up the terms with two radicals in them. Notice that, in this case, the answer has no radicals. That will happen on occasion so don’t get excited ab