document.write( "Question 383640: Please help me with this problem: Rationalize the denominator. Assume that all expressions under the radical represent positive numbers. 4√7/√5-√3 \n" ); document.write( "

Algebra.Com's Answer #271703 by jsmallt9(3758) $\"\"$ $\"About$

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$\"4sqrt%287%29%2F%28sqrt%285%29-sqrt%283%29%29\"$
\n" ); document.write( "Your denominator has two terms. Rationalizing a two term denominator uses the pattern: $\"%28a%2Bb%29%28a-b%29+=+a%5E2-b%5E2\"$ . The pattern shows us how a binomial, (a+b) or (a-b), can be turned into an expression of perfect squares, $\"a%5E2-b%5E2\"$ !

\n" ); document.write( "Your denominator has a minus between the two terms. So it will plat the role of (a-b) with \"a\" being $\"sqrt%285%29\"$ and \"b\" being $\"sqrt%283%29\"$ . To rationalize this denominator we will multiply the numerator and denominator by (a+b):
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\n" ); document.write( "In the numerator we will use the distributive Property to multiply. In the denominator we already know we will get $\"a%5E2=b%5E2\"$ :
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\n" ); document.write( "which simplifies as follows:
\n" ); document.write( " $\"%284sqrt%2835%29%2B4sqrt%2821%29%29%2F%285-3%29\"$
\n" ); document.write( "You can already see that the denominator is rational.
\n" ); document.write( " $\"%284sqrt%2835%29%2B4sqrt%2821%29%29%2F2\"$
\n" ); document.write( "We can reduce this fraction by factoring out a 2 in the numerator:
\n" ); document.write( " $\"%282%282sqrt%2835%29%2B2sqrt%2821%29%29%29%2F2\"$
\n" ); document.write( " $\"%28cross%282%29%282sqrt%2835%29%2B2sqrt%2821%29%29%29%2Fcross%282%29\"$
\n" ); document.write( " $\"2sqrt%2835%29%2B2sqrt%2821%29\"$
\n" ); document.write( "Not only did we rationalize the denominator but we eliminated the fraction entirely! \n" ); document.write( "

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