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A Special Case of Rationalizing / Radicals & Exponents / Radicals & Domains (page 7 of 7) Sections: Square roots, More simplification / Multiplication, Adding (and subtracting) square roots, Conjugates / Dividing by square roots, Rationalizing denominators, HigherIndex Roots, A special case of rationalizing / Radicals & exponents / Radicals & domains A Special Case of Rationalizing If your class has covered the formulas for factoring the sums and differences of cubes, then you might encounter a special case of rationalizing denominators. The reasoning and methodology are similar to the "difference of squares" conjugate process for square roots.
I would like to get rid of the cube root, but multiplying by the conjugate won't help much: But I can "create" a sum of cubes, just as using the conjugate allowed me to create a difference of squares earlier. Using the fact that a^{3} + b^{3} = (a + b)(a^{2} – ab + b^{2}), and letting a = 1 and b equal the cube root of 4, I get: If I multiply, top and bottom, by the second factor in the sumofcubes formula, then the denominator will simplify with no radicals: Naturally, if the sign in the middle of the original denominator had been a "minus", I'd have applied the "difference of cubes" formula to do the rationalization. This sort of "rationalize the denominator" exercise almost never comes up. But if you see this in your homework, expect one of these on your next test. Radicals Expressed With Exponents Radicals can be expressed as fractional exponents. Whatever is the index of the radical becomes the denominator of the fractional power. For instance: The second root became a onehalf power. A cube root would be a onethird power, a fourth root would be a onefourth power, and so forth.This conversion process will matter a lot more once you get to calculus. For now, it allows you to simplify some expressions that you might otherwise not have been able to. Copyright © Elizabeth Stapel 19992011 All Rights Reserved
I will convert the radicals to exponential expressions, and then apply exponent rules to combine the factors:
A Few Other Considerations Usually, we cannot have a negative inside a square root. (The exception is for "imaginary" numbers. If you haven't done the number "i" yet, then you haven't done imaginaries.) So, for instance, is not possible. Do not try to say something like " ", because it's not true: . You must have a positive inside the square root. This can be important for defining and graphing functions.
The fact that I have the expression x – 2 inside a square root requires that x – 2 be zero or greater, so I must have x – 2 > 0. Solving, I get: domain: x > 2 On the other hand, you CAN have a negative inside a cube root (or any other odd root). For instance: ...because (–2)^{3} = –8.
For , there is NO RESTRICTION on the value of x, because x – 2 is welcome to be negative inside a cube root. Then the domain is: domain: all x << Previous Top  1  2  3  4  5  6  7  Return to Index



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