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Radicals. Laws. Simplification. Reduction of the index. Rationalization of the denominator.


Def. Radical. An expression of the form ole.gif denoting the principal n-th root of a. The positive integer n is the index, or order, of the radical and the number a is the radicand. The index is omitted if n = 2.


The laws for radicals are obtained directly from the laws for exponents by means of the definition


             ole1.gif


Laws of Radicals. If n is even, assume a, b ≥ 0.


ole2.gif



Simplification of radicals. It is important to reduce a radical to its simplest form through use of the following operations.


1. Removal of perfect n-th powers from a radicand. Any radical of order n should be simplified by removing all perfect n-th powers from under the radical sign using the rule ole3.gif .


Examples.


ole4.gif



2. Reduction of the index of the radical.


Examples.


ole5.gif


In the first example the index was reduced from 4 to 2 and in the second example it was reduced from 6 to 3. We note that the process involves converting to exponential notation and then converting back.


3. Rationalization of the denominator. Fractions may be removed from under a radical sign by rationalizing the denominator. To rationalize the denominator of a radical of order n, multiply the numerator and denominator of the radicand by such a quantity as will make the denominator a perfect n-th power and then remove the denominator from under the radical sign.


Examples.


ole6.gif




A radical is said to be in simplest form if

 

1)        all perfect n-th powers have been removed from the radical

2)        the index of the radical is as small as possible

3)        no fractions are present in the radicand i.e. the denominator has been rationalized



Def. Similar radicals. Radicals which, on being reduced to simplest form, have the same index and radicand.


ole7.gif



Addition and subtraction of radicals. Before addition or subtraction of radicals it is important to reduce them to simplest form. Like radicals can then be added or subtracted in the same way as other like terms.



Example.


             ole8.gif





Multiplication of radicals.


1) To multiply two or more radicals having the same index use ole9.gif .


Examples.


ole10.gif



2) To multiply radicals with different indices use fractional exponents and the laws of exponents.


Example.


             ole11.gif




Division of radicals.


1) To divide two radicals having the same index use


             ole12.gif

and simplify.



Example.


             ole13.gif



2) To divide radicals with different indices use fractional exponents and the laws of exponents.



Example.


             ole14.gif



 


Rationalization of the denominator in fractions with binomial irrational denominators. In one form of fraction, the denominator is a binomial in which one term is a square root of a rational number or expression and the other term is of the same form or is rational i.e. the denominator has the form ole15.gif . In such a case, rationalize the fraction by multiplying the numerator and denominator by the conjugate of the denominator - where the conjugate of ole16.gif and the conjugate of ole17.gif .



Example.


             ole18.gif

 





References

 Hawks, Luby, Touton. Second-Year Algebra

 Murray R. Spiegel. College Algebra

 Raymond W. Brink. A First Year of College Mathematics


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