document.write( "Question 378084: Simplify.
\n" ); document.write( "4√3 - 1 / 2√3 + 2 \n" ); document.write( "
Algebra.Com's Answer #268656 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
\"4sqrt%283%29+-+1%2F2sqrt%283%29+%2B+2\"
\n" ); document.write( "If your problem is \"4sqrt%283%29+-+1%2F%282sqrt%283%29+%2B+2%29\" then a) put multiple term numerators and denominators in parentheses; and b) see the solution below.

\n" ); document.write( "First we'll rationalize the denominator of the middle term. Since there is just single term in the denominator this is fairly easy. We just multiply both the numerator and denominator by \"sqrt%283%29\":
\n" ); document.write( "\"4sqrt%283%29+-+%281%2F2sqrt%283%29%29%28sqrt%283%29%2Fsqrt%283%29%29+%2B+2\"
\n" ); document.write( "giving us:
\n" ); document.write( "\"4sqrt%283%29+-+%281%2Asqrt%283%29%29%2F%282%2Asqrt%283%29%2Asqrt%283%29%29+%2B+2\"
\n" ); document.write( "which simplifies as follows:
\n" ); document.write( "\"4sqrt%283%29+-+sqrt%283%29%2F%282%2A3%29+%2B+2\"
\n" ); document.write( "\"4sqrt%283%29+-+sqrt%283%29%2F6+%2B+2\"
\n" ); document.write( "Then we can make all the denominators the same (so we can add and subtract):
\n" ); document.write( "\"24sqrt%283%29%2F6+-+sqrt%283%29%2F6+%2B+12%2F6\"
\n" ); document.write( "Adding and subtracting we get:
\n" ); document.write( "\"%2824sqrt%283%29+-+sqrt%283%29+%2B+12%29%2F6\"
\n" ); document.write( "The first two terms are like terms and can be subtracted:
\n" ); document.write( "\"%2823sqrt%283%29+%2B+12%29%2F6\"
\n" ); document.write( "Either this or \"23sqrt%283%29%2F6+%2B+2\" is the simplified answer.

\n" ); document.write( "If the problem was:
\n" ); document.write( "\"4sqrt%283%29+-+1%2F%282sqrt%283%29+%2B+2%29\"
\n" ); document.write( "Just like above we want to rationalize the second denominator. And just like above we will multiply the numerator and denominator to do so. The difference is that with a denominator of two terms it is not as easy to figure out what to multiply by. Our goal is to eliminate the square root in the denominator. So we have to figure out: \"What can we multiply \"%282sqrt%283%29+%2B+2%29\" that will cause the square root to disappear?\" With two terms, the key is the pattern \"%28a%2Bb%29%28a-b%29+=+a%5E2+-+b%5E2\". This shows us how to take a two-term expression, like a+b or a-b, and multiply it and get an expression of nothing but perfect square terms. Since \"%282sqrt%283%29+%2B+2%29\" has a plus sign between the terms, we will use \"%282sqrt%283%29+-+2%29\":
\n" ); document.write( "
\n" ); document.write( "We'll use the Distributive Property to multiply the numerators. With the denominators we will just use the pattern:
\n" ); document.write( "\"4sqrt%283%29+-+%282sqrt%283%29+-+2%29%2F%28%282sqrt%283%29%29%5E2+-+2%5E2%29\"
\n" ); document.write( "which simplifies as follows:
\n" ); document.write( "\"4sqrt%283%29+-+%282sqrt%283%29+-+2%29%2F%284%2A3+-+4%29\"
\n" ); document.write( "\"4sqrt%283%29+-+%282sqrt%283%29+-+2%29%2F%2812+-+4%29\"
\n" ); document.write( "\"4sqrt%283%29+-+%282sqrt%283%29+-+2%29%2F8\"
\n" ); document.write( "The denominator is now rational. We can proceed to subtracting. First we need common denominators:
\n" ); document.write( "\"32sqrt%283%29%2F8+-+%282sqrt%283%29+-+2%29%2F8\"
\n" ); document.write( "Subtracting we get:
\n" ); document.write( "\"%2832sqrt%283%29+-+2sqrt%283%29+%2B+2%29%2F8\"
\n" ); document.write( "The first two terms in the numerator are like terms:
\n" ); document.write( "\"%2830sqrt%283%29+%2B+2%29%2F8\"
\n" ); document.write( "If we factor out a 2 in the numerator this fraction will reduce:
\n" ); document.write( "\"%282%2815sqrt%283%29+%2B+1%29%29%2F%282%2A4%29\"
\n" ); document.write( "\"%28cross%282%29%2815sqrt%283%29+%2B+1%29%29%2F%28cross%282%29%2A4%29\"
\n" ); document.write( "leaving\"
\n" ); document.write( "\"%2815sqrt%283%29+%2B+1%29%2F4\"
\n" ); document.write( "This expression or \"15sqrt%283%29%2F4+%2B+1%2F4\" is the simplified expression. \n" ); document.write( "
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