Rationalizing Denominators

Transcription

1 Solver9 Rationalizing Denominators Get that pesky radical OUT of my denominator John Reece 7//00

2 What does Rationalize the Denominator mean? It means to change a fraction, without changing its value, so that any irrational or complex numbers in the denominator of the fraction become rational numbers. For example, is a fraction with an irrational number in the denominator, whereas is an equivalent number, but with a rational number denominator. Before the invention of readily accessible calculators, when people did calculations by hand, sometimes supported by logarithm tables or by a slide rule, it was much easier to calculate the numerical approximation of a number such as than it would be to calculate the value of. mathematicians have decreed that numbers with rational denominators are simpler than numbers with irrational denominators. Hence, by convention, mathematicians have, in general, decreed that numbers with rational denominators are simpler than those with irrational denominators. Notice that the convention is qualified with the phrase in general. That is because there are situations, albeit infrequent, when you may not want to rationalize the denominator, or perhaps even rationalize a numerator at the expense of an irrational denominator. However, in the vast majority of cases, you will be either asked or required to perform the process of rationalizing denominators.

3 Rationalizing a monomial denominator. The simplest situation is that illustrated in the example given above where the denominator consists of a single radical term. The process of rationalizing the denominator makes use of the definition of the radical symbol and the concept of the multiplicative identity. Definition of the Radical Symbol: Multiplicative Inverse: y x y x a, a 0, such that a a a Multiplicative Identity: a, I such that a I a So how do we apply these ideas to a number that looks like: and alter it so that it retains its current value, yet has a rational number for a denominator? by The definition of the radical symbol tells us that if we multiply the denominator that we will end up with, a rational number, in our denominator. The idea of a multiplicative inverse tells us that if we multiply the value that we need to get rid of the radical in the denominator by we will have a multiplier of, and then the idea of the multiplicative identity guarantees that using this multiplier will not change the value of our original fraction. is the required multiplication simply proceed as in multiplying any other fractions, numerator times numerator and denominator times denominator:

4 And the task is complete. Rationalizing a binomial denominator. This is a bit trickier involving a fractional expression where the denominator consists of two terms at least one of which is a radical. Example: If we try the method of the previous section, that is to multiply by a factor of the denominator divided by itself, we won t get any closer to our goal of eliminating the radical in our original denominator. Here s why: We still have a radical in the denominator! Obviously, we need a different strategy. The strategy to solve this type of problem involves the concept of a conjugate binomial. The conjugate of any binomial is simply the binomial with the sign between the two terms reversed. For example, a band a bare conjugates. The important property of conjugates is that when you multiply them, you get the difference of two squares. a b a b a b Ah ha! This looks promising. If we multiply our two-term denominator by its conjugate, we can get rid of the original radical without introducing a new one. 4 Much better. Of course, we need to consider the multiplicative identity and multiply our original fraction by, but this time in terms of the conjugate of the denominator divided by itself: Whoa! We got rid of the denominator altogether swwwwweet!

5 Rationalizing a monomial denominator with index >. Ok, we have handled monomial and binomial denominators with square roots, but what about higher order roots, i.e. cube roots or 4 th roots and such? 4 5 If we simply multiply by the denominator divided by the denominator, we are out of luck again means to take the 4 th root of something. and we still have a radical. Let s look at what it 4 4 y x y x and bye- 4 But notice that y y y, so that means if I multiply bye you nasty old radical. Which leads us to the solution: And that s all I have to say about that. Rationalizing a complex denominator. A complex denominator is one that is of the form a biwhere a and b are real numbers and i is the imaginary number defined by i. Example: 5 i Again, we need to return to our concept of the conjugate. Complex conjugates are formed the same way any other binomial conjugates are formed; just change the sign on the imaginary part coefficient. For the example: 5 i 5 0i 5 0i i i 9 ( )(4) The only trick is to remember thati.

6 Conclusion: This tutorial reference deals with the sorts of situations that are typically encountered in Algebra I and II, Trigonometry, Pre-Calculus, and most, if not all, of Calculus, but is not intended to be an exhaustive treatment of the subject. If you have a situation not covered in this tutorial, by all means ask a question on the Q&A board. Solver9 reecejo on Google s mail service.