This week in Pre Cal, we learned more about Roots and Radicals, whereas there’s multiplication and divisions involved. Furthermore, we learned about rationalizing a denominator.
Primarily, we learned how to simplify radical expressions with variable radicands as well as some interesting tricks that makes the simplifying easier to do so.
I will be using this example of a radical expression to simplify, as well as to identify the values of each variable:
First step is to observe the radical expression since we need to know what method we are going to use, and as you can see, the binomial radical expression have same terms except that the signs are different (first term has a negative [minus] sign, while the second term has a positive sign [plus]) – which means these two binomials are CONJUGATES of each other. Furthermore, we can say that these are difference of squares.
Second step is to do the FOIL method to simplify. Since, these are difference of squares, our distributing will be easier because apparently, after we distribute the terms, we end up with the middle terms cancelling each other out. So, to make our distribution simpler when we have difference of squares, we multiply the First terms and the Last terms only.
Third step is to simplify further by cancelling the exponent inside the radical as well as the root sign because:
Last step is to identify the values of the variables (x and y) of this expression. Since this is an even root sign, the value of the radicands has to be positive otherwise, when the radicand is negative, it would result as undefined/not defined because when you multiply a number itself twice, it would always be positive even when there’s a negative sign with the number. Therefore, the values of the variable radicands needs to be restricted in order for its value to be positive when evaluated. So, it would be (x is greater or equal to zero) and (y is greater or equal to zero).
Here’s the full image: