document.write( "Question 201791: sqrt (56ab^3) / sqrt (7a) \n" ); document.write( "
Algebra.Com's Answer #152035 by jsmallt9(3758)\"\" \"About 
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To simplify \"sqrt+%2856ab%5E3%29+%2F+sqrt+%287a%29\" we will need a couple of basic properties of square roots:
\n" ); document.write( "\"sqrt%28x%2Fy%29+=+sqrt%28x%29%2Fsqrt%28y%29\" and
\n" ); document.write( "\"sqrt%28x%2Ay%29+=+sqrt%28x%29%2Asqrt%28y%29\"
\n" ); document.write( "We can use the first property to combine your fraction of square roots into a square root of a fraction:
\n" ); document.write( "\"sqrt+%2856ab%5E3%29+%2F+sqrt+%287a%29+=+sqrt%28%2856ab%5E3%29+%2F+%287a%29%29\"
\n" ); document.write( "We do this because we can cancel factors and reduce the fraction:
\n" ); document.write( "\"sqrt%28%2856ab%5E3%29+%2F+%287a%29%29+=+sqrt%28%287%2A8%2Aa%2Ab%5E3%29%2F%287a%29%29\"
\n" ); document.write( "The 7's and the a's cancel leaving:
\n" ); document.write( "\"sqrt%288b%5E3%29\"
\n" ); document.write( "Now we factor out as many perfect squares as we can find:
\n" ); document.write( "\"sqrt%284%2A2%2Ab%5E2%2Ab%29\"
\n" ); document.write( "We can use the Commutative property to rearrange the order:
\n" ); document.write( "\"sqrt%284%2Ab%5E2%2A2b%29\"
\n" ); document.write( "And now we can use the second property to separate out the perfect square factors:
\n" ); document.write( "\"sqrt%284%29%2Asqrt%28b%5E2%29%2Asqrt%282b%29\"
\n" ); document.write( "The \"sqrt%284%29+=+2\".
\n" ); document.write( "And normally we would say that \"sqrt%28b%5E2%29+=+abs%28b%29\" so that we can guarantee a non-negative value for the value of the square root. But if we look back at the original expression, we can see that:
  1. \"a\" cannot be zero because it would make the denominator zero.
  2. \"a\" cannot be negative because the radicand (the number inside the square root) must not be negative
  3. So \"a\" must be positive.
  4. Since \"a\" is positive \"b%5E3\" must be zero or positive so that the radicand in the numerator is not negative.
  5. Since \"b%5E3\" is zero or positive then \"b\" must be zero or positive.
  6. If \"b\" is zero or positive, then \"abs%28b%29+=+b\"

\n" ); document.write( "So we do not need |b| to guarantee a non-negative square root. We can use just plain \"b\".
\n" ); document.write( "Substituting into our expression we get:
\n" ); document.write( "\"2%2Ab%2Asqrt%282b%29\"
\n" ); document.write( "or simply
\n" ); document.write( "\"2b%2Asqrt%282b%29\" \n" ); document.write( "
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