Question 374497
{{{sqrt(7x^19*y^7)/sqrt(2x^2*y^2)}}}
Simplifying square roots involves making sure that<ul><li>No square root has a perfect square factor</li><li>No square root has a fraction in it</li><li>No fractions have a square root in its denominator</li></ul>When the expression does not involve fractions only the first item needs to be addressed.<br>
Any valid math which achieves these goals will work. With an expression like yours, a fraction of square roots, what I like to do is:<ol><li>Use the property of radicals, {{{root(a, p)/root(a, q) = root(a, p/q)}}}, to turn the expression into a square root of a fraction.</li><li>Reduce the fraction as much as possible.</li><li>If there is still a fraction after the reducing, <ol><li>turn its denominator into a perfect square.</li><li>Use the same property as step #1, only in reverse, to turn the expression back into a fraction of square roots</li><li>Simplify the denominator (which should be the square root of a perfect square)</li></ol></li><li>If there is still a square root at this point, use another property of radicals, {{{root(a, p)*root(a, q) = root(a, p*q)}}}, to separate out perfect square factors, if any.</li></ol>Let's see how this works:
1) Square root of a fraction
{{{sqrt((7x^19*y^7)/(2x^2*y^2))}}}
2) Reduce the fraction
{{{sqrt((7x^2*x^17y^2*y^5)/(2x^2*y^2))}}}
{{{sqrt((7cross(x^2)*x^17*cross(y^2)*y^5)/(2cross(x^2)*cross(y^2)))}}}
{{{sqrt((7*x^17*y^5)/2)}}}
3.1) Make the remaining denominator a perfect square:
{{{sqrt(((7*x^17*y^5)/2)*(2/2))}}}
{{{sqrt((14*x^17*y^5)/4)}}}
3.2) Rewrite as a fraction of square roots:
{{{sqrt(14*x^17*y^5)/sqrt(4)}}}
3.3) Simplify the denominator
{{{sqrt(14*x^17*y^5)/2}}}
4) Simplify the remaining square root
{{{sqrt(14*x^2*x^2*x^2*x^2*x^2*x^2*x^2*x^2*x*y^2*y^2*y)/2}}}
(Use the Commutative property to rearrange the order so all the perfect squares are together.)
{{{sqrt(x^2*x^2*x^2*x^2*x^2*x^2*x^2*x^2*x*y^2*y^2*14xy)/2}}}
{{{sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(x^2)*sqrt(y^2)*sqrt(y^2)*sqrt(14xy)/2}}}
{{{(x*x*x*x*x*x*x*x*y*y*sqrt(14xy))/2}}}
{{{(x^8*y^2*sqrt(14xy))/2}}}
This is the simplified expression. (It meets all three conditions described above.)<br>
Note: Square roots, unless they have a "-" in front of them, are supposed to be positive (or zero). When simplifying an expression of square roots, you sometimes need to use absolute value to ensure your simplified expression is positive (or zero), too. In this problem, however, the {{{x^8*y^2}}} cannot be negative because of the even exponents. So we do not need any absolute values.