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Why is it important to simplify radical expressions before adding or subtracting? How is adding radical expressions similar to adding polynomial expressions? How is it different? Provide a radical expression for the assignment to simplify.
|(a) It is important to simplify radical expressions before adding/subtracting because only like radicals can be added / subtracted and the results expressed as |
|single radical. |
|(b) Adding radical expressions is similar to adding polynomials, in that the same laws are followed in the process. |
|It is different in the sense that whereas in polynomials we have integer powers of the variables that are being added or subtracted, in case of radical |
|expressions, we have powers of the variables as fractions. |
|(c) Examples: (a) Simplify ((32x^3) (b) Add ((4x) and ((9x) |
|[Answers: (a) 4x ((2x) (b) 5 (x |
|(d) Consider the second example above. Had we not simplified the radicals as 2 √x and 3 √x, we would not have been able to keep the final answer as 5 √x. (We |
|would have retained the answer as just √4x and √9x. |
What is the index of a radical? When working with radicals, can the radicand be negative when the index is odd? Can it be negative when the index is even? Provide an example for a radical involving a negative radicand for others to simplify.
|(a) Consider √x. This is "second root" of x. Here 2 is called the index of the radical. |
| Similarly, for 3√x, which is the "third root" of x, 3 is the index of the radical. |
| In general, for the radical n√x, n is called the index of the radical. |
| (b) Yes, it can be. For example, 3√-8 is valid and the answer is -2. |
| (c) No, it can't be. For example, √-4 is not real. It is a complex number. Similarly, 4√-16 is not real. |
| (d) An example is: Simplify 3√-27. |
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