document.write( "Question 203596: √8(2√6-4√10) \n" ); document.write( "

Algebra.Com's Answer #153619 by jsmallt9(3758) $\"\"$ $\"About$

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We'll need one of the properties of square roots: $\"sqrt%28x%29%2Asqrt%28y%29+=+sqrt%28x%2Ay%29\"$ . This property can be used to multiply square roots. And it can also be used to split the square root of a product into the product of square roots. We will use both.

\n" ); document.write( " $\"sqrt%288%29%282%2Asqrt%286%29+-+4%2Asqrt%2810%29%29\"$
\n" ); document.write( "We can use the Distributive Property to multiply these:
\n" ); document.write( " $\"sqrt%288%29%2A2%2Asqrt%286%29+-+sqrt%288%29%2A4%2Asqrt%2810%29%29\"$
\n" ); document.write( "Then the Commutative Property and Associative Property to reorder and regroup each term:
\n" ); document.write( " $\"2%28sqrt%288%29%2Asqrt%286%29%29+-+4%28sqrt%288%29%2Asqrt%2810%29%29\"$
\n" ); document.write( "Then the square root property to multiply the square roots:
\n" ); document.write( " $\"2sqrt%2848%29+-+4sqrt%2880%29\"$
\n" ); document.write( "It appears that there is nothing further that can be done. If we had fractions we would reduce them, if possible. And it is the same with square roots. Simplify them, if possible, before stopping. We simplify square roots by finding perfect square factors in the radicand (the expression inside the square root), if possible. Both 48 and 80 have perfect square factors. By coincidence it is the same perfect square: 16. So we can simplify by:

Factoring the radicand where as many factors as possible are perfect squares (other than 1 which is a factor of everything).
Use the square root property (above) to separate the factors into different square roots.
Simplify the square roots of the perfect squares.
Simplify what is left, including adding or subtracting \"like\" square roots, if any.

\n" ); document.write( "So we will start to simplify $\"2sqrt%2848%29+-+4sqrt%2880%29\"$
\n" ); document.write( "1) Find as many perfect square factors as possible:
\n" ); document.write( " $\"2sqrt%2848%29+-+4sqrt%2880%29+=+2sqrt%2816%2A3%29+-+4sqrt%2816%2A5%29\"$
\n" ); document.write( "2) Separate the perfect square factors into their own square roots, using the property above:
\n" ); document.write( "

\n" ); document.write( "3) Simplify the square roots of the perfect squares:
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\n" ); document.write( "4) Simplify
\n" ); document.write( " $\"2%2A4%2Asqrt%283%29+-+4%2A4%2Asqrt%285%29+=+8sqrt%283%29+-+16sqrt%285%29\"$
\n" ); document.write( "If these were \"like\" square roots we would then subtract them. (\"Like\" square roots are a lot like \"like\" terms when adding with variables. We can add 2x + 3x and get 5x. We can add $\"2sqrt%2815%29+%2B+3sqrt%2815%29+=+5sqrt%2815%29\"$ . But we cannot add 2x + 3y because they are not \"like\" terms.) But we cannot subtract $\"8sqrt%283%29+-+16sqrt%285%29\"$ because they are not \"like\" square roots.)

\n" ); document.write( "So $\"sqrt%288%29%282%2Asqrt%286%29+-+4%2Asqrt%2810%29%29+=+8sqrt%283%29+-+16sqrt%285%29\"$ in simplest form.\r
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