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Thabang from Lesotho writes, "how do we rationalize a denominator consisting of a cube root with another constant added to it or subtracted from it?"
Good morning, Thabang, and thank you for your question. This is actually something I don't remember ever seeing before, so I had to give it some thought before answering.
What you're looking for is, how do we rationalize the denominator, if the denominator is something like "The cube root of three, plus two" or "the cube root of three, minus two"?
In order to solve this, it's important to remember two factoring rules you may have learned in an Algebra class:
x3 + y3 = (x + y)(x2 - xy + y2)
x3 - y3 = (x - y)(x2 + xy + y2)
Let's say your denominator is the cube root of three, plus two. Then I'm going to do the following substitutions:
Let x = the cube root of three, let y = 2.
Now your denominator is x + y, and if you multiply the numerator and denominator of the fraction by (x2 - xy + y2), you will have turned the denominator into x3 + y3 = 3 + 8 = 11, which is rational.
That was using the first factoring rule shown above. If the denominator had a subtraction (the cube root of three, minus two), we'd just use the second factoring rule, and multiply by (x2 + xy + y2).
Thanks again for asking, Thabang.