document.write( "Question 219955: What is 3 divided by 5 + the square root of 2? I don't know where to start. \n" ); document.write( "

Algebra.Com's Answer #165315 by jsmallt9(3758) $\"\"$ $\"About$

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(NOTE: One of the answers to this problem provided by someone else is incorrect.)
$\"3%2F%285%2Bsqrt%282%29%29\"$
\n" ); document.write( "With the $\"sqrt%282%29\"$ you have an irrational number in the denominator. So the problem is to rationalize the denominator. IOW we need to eliminate any square roots in the denominator.

\n" ); document.write( "If the denominator was just $\"sqrt%282%29\"$ we could simply multiply the numerator and denominator by $\"sqrt%282%29\"$ and this would get rid of the square root in the denominator. For example: $\"7%2Fsqrt%282%29+=+%287%2Fsqrt%282%29%29%28sqrt%282%29%2Fsqrt%282%29%29+=+7sqrt%282%29%2F2\"$

\n" ); document.write( "But this will not work on your fraction beacause of the additional term in the denominator:

So we need to do something different.

\n" ); document.write( "THe \"trick\" is to understand the pattern: $\"%28a%2Bb%29%28a-b%29+=+a%5E2+-+b%5E2\"$ . \"a+b\" and \"a-b\" are called conjugates. And the pattern tells us if you multiply conjugates you get the difference of the squares of the two terms in the conjugates. This expression of squared terms is what we need to rationalize the denominator.

\n" ); document.write( "So what we will do is multiply the numerator and denominator by the conjugate of $\"5%2Bsqrt%282%29\"$ which is $\"5-sqrt%282%29\"$ :
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