Radicals  Rationalize Denominators


 Bartholomew Watkins
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1 8. Radicals  Rationalize Denominators Objective: Rationalize the denominators of radical expressions. It is considered bad practice to have a radical in the denominator of a fraction. When this happens we multiply the numerator and denominator by the same thing in order to clear the radical. In the lesson on dividing radicals we talked about how this was done with monomials. Here we will look at how this is done with binomials. If the binomial is in the numerator the process to rationalize the denominator is essentially the same as with monomials. The only difference is we will have to distribute in the numerator. Example 1. ( ) ( 9) Want to clear 6 in denominator, multiply by 6 6 We will distribute the 6 through the numerator 1
2 Simplify radicals in numerator, multiply out denominator Take square root where possible Reduce by dividing each term by It is important to remember that when reducing the fraction we cannot reduce with just the and 1 or just the 9 and 1. When we have addition or subtraction in the numerator or denominator we must divide all terms by the same number. The problem can often be made easier if we first simplify any radicals in the problem. 0x 1x 18x Simplify radicals by finding perfect squares x x Simplify roots, divide exponents by. 9 x x x x x ( (x x x ) x Multiply coefficients x x x Multiplying numerator and denominator by x x ) x x Distribute through numerator x 10x x x 6x x 10 x 6x 6x Simplify roots in numerator, multiply coefficients in denominator Reduce, dividing each term by x
3 x 10 6x x As we are rationalizing it will always be important to constantly check our problem to see if it can be simplified more. We ask ourselves, can the fraction be reduced? Can the radicals be simplified? These steps may happen several times on our way to the solution. If the binomial occurs in the denominator we will have to use a different strategy to clear the radical. Consider, if we were to multiply the denominator by we would have to distribute it and we would end up with. We have not cleared the radical, only moved it to another part of the denominator. So our current method will not work. Instead we will use what is called a conjugate. A conjugate is made up of the same terms, with the opposite sign in the middle. So for our example with in the denominator, the conjugate would be +. The advantage of a conjugate is when we multiply them together we have ( )( + ), which is a sum and a difference. We know when we multiply these we get a difference of squares. Squaring and, with subtraction in the middle gives the product =. Our answer when multiplying conjugates will no longer have a square root. This is exactly what we want. Example. ( ) Multiply numerator and denominator by conjugate Distribute numerator, difference of squares in denominator Simplify denoinator Reduce by dividing all terms by In the previous example, we could have reduced by dividng by, giving the solution +, both answers are correct. 11 Example. 1 Multiply by conjugate, +
4 ( ) Distribute numerator, denominator is difference of squares Simplify radicals in numerator, subtract in denominator Take square roots where possible Example. x x ( x Multiply by conjugate, + x x + x + x ) Distribute numerator, denominator is difference of squares 8 x + 1x Simplify radicals where possible 16 x 8 x + x 1 16 x The same process can be used when there is a binomial in the numerator and denominator. We just need to remember to FOIL out the numerator. Example. ( ) + + Multiply by conjugate, + FOIL in numerator, denominator is difference of squares Simplify denominator Divide each term by 1
5 Example 6. ( ) Multiply by the conjugate, FOIL numerator, denominator is difference of squares Multiply in denominator Subtract in denominator The same process is used when we have variables Example 7. x x + x Multiply by the conjugate, x + x x x ( ) x x + x x + x x x x + x FOIL in numerator, denominator is difference of squares 1x x +x 6x +x x + 1x x x 1x x + x x x + x x x 1x x +x x x +x x Simplify radicals Divide each term by x World View Note: During the th century BC in India, Aryabhata published a treatise on astronomy. His work included a method for finding the square root of numbers that have many digits. Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution.0 Unported License. (
6 8. Practice  Rationalize Denominators Simplify. 1) + 9 ) + ) 1 7) ) + 9 ) 16 6) 8) ) + 10) + 11) + 1) 1) 1) 1) + 16) + 17) 19) 1) ) ) ab a b a a+ ab a + b 7) ) a b a+ b 18) 0) + 1 ) ) 6) ) 0) a+ ab a + b + 1 a b a + b 1) 6 ) ab a b b a ) ) a b a b b a ) 6)
7 7) + 8) + + Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution.0 Unported License. ( 7
8 8. 1) + ) + 1 ) + 1 ) ) ) ) 0 8) 18 9) 1 10) ) 10 1) + 1) ) 1) 16) 17) 1+ 18) ) 1 Answers  Rationalize Denominators 0) + 1) ) ) a ) ) a 6) 1 7) + 6 8) ) a a b + b a b 0) a b 1) + ) a b + b a a b ) a b + b a ab ) ) 1+ 6) ) ) Beginning and Intermediate Algebra by Tyler Wallace is licensed under a Creative Commons Attribution.0 Unported License. ( 8
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