document.write( "Question 356972: Multiply. Assume variables represent nonnegative real numbers
\n" ); document.write( "1. sqrt(3)*(2sqrt(15)-3sqrt(4))
\n" ); document.write( "2. (3+sqrt(11))*(3-sqrt(11))
\n" ); document.write( "3. (sqrt(5)-sqrt(3))^2 \n" ); document.write( "

Algebra.Com's Answer #254950 by jsmallt9(3758) $\"\"$ $\"About$

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1. $\"sqrt%283%29%2A%282sqrt%2815%29-3sqrt%284%29%29\"$
\n" ); document.write( "Since $\"sqrt%284%29+=+2\"$ , we can start by substituting:
\n" ); document.write( " $\"sqrt%283%29%2A%282sqrt%2815%29-3%282%29%29\"$
\n" ); document.write( "or
\n" ); document.write( " $\"sqrt%283%29%2A%282sqrt%2815%29-6%29\"$
\n" ); document.write( "Now we can multiply, using the Distributive Property:
\n" ); document.write( " $\"sqrt%283%29%2A2sqrt%2815%29-sqrt%283%29%2A6\"$
\n" ); document.write( "In the first product we use the property of radicals, $\"root%28a%2C+p%29%2Aroot%28a%2C+q%29+=+root%28a%2C+p%2Aq%29\"$ , to multiply the two square roots. In the second factor we just use the Commutative Property to move the 6 in front:
\n" ); document.write( " $\"2sqrt%283%2A15%29-6sqrt%283%29\"$
\n" ); document.write( " $\"2sqrt%2845%29-6sqrt%283%29\"$
\n" ); document.write( "We cannot subtract these two terms because they are not like terms. But we can simplify the first square root. (It has a perfect square factor: 9)
\n" ); document.write( " $\"2sqrt%289%2A5%29-6sqrt%283%29\"$
\n" ); document.write( "Using the same property of radicals as above (only \"in reverse\") we can split these two square roots:
\n" ); document.write( " $\"2sqrt%289%29%2Asqrt%285%29-6sqrt%283%29\"$
\n" ); document.write( "Since $\"sqrt%289%29+=+3\"$ :
\n" ); document.write( " $\"2%283%29%2Asqrt%285%29-6sqrt%283%29\"$
\n" ); document.write( "or
\n" ); document.write( " $\"6sqrt%285%29-6sqrt%283%29\"$

\n" ); document.write( "2. $\"%283%2Bsqrt%2811%29%29%2A%283-sqrt%2811%29%29\"$
\n" ); document.write( "We can use FOIL to multiply this. Or we can take advantage of the difference of squares pattern, $\"%28a%2Bb%29%28a-b%29+=+a%5E2+-+b%5E2\"$ , with \"a\" being 3 and \"b\" being $\"sqrt%2811%29\"$ :
\n" ); document.write( " $\"%283%29%5E2+-+%28sqrt%2811%29%29%5E2\"$
\n" ); document.write( "9 - 11
\n" ); document.write( "-2

\n" ); document.write( "3. $\"%28sqrt%285%29-sqrt%283%29%29%5E2\"$
\n" ); document.write( "For this we can use FOIL. Or we can use a pattern, $\"%28a-b%29%5E2+=+a%5E2+-2ab+%2B+b%5E2\"$ with \"a\" being $\"sqrt%285%29\"$ and \"b\" being $\"sqrt%283%29\"$ :
\n" ); document.write( " $\"%28sqrt%285%29%29%5E2+-+2%28sqrt%285%29%29%28sqrt%283%29%29+%2B+%28sqrt%283%29%29%5E2\"$
\n" ); document.write( " $\"5+-2sqrt%2815%29+%2B+3\"$
\n" ); document.write( " $\"8+-+2sqrt%2815%29\"$

\n" ); document.write( "Note: Even if you use FOIL on #2 and #3, you still end up with the same answers as we did using patterns. \n" ); document.write( "

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