Question 404197
{{{sqrt(x^3)/sqrt(12)}}}
While there are other ways, the way I like to simplify radical over radical fractions is:<ol><li>Use a property of radicals, {{{root(a, p)/root(a, q) = root(a, p/q)}}}, to rewrite the fraction of radicals into the radical of a fraction.</li><li>Reduce the fraction inside the radical, if possible.</li><li>Multiply the numerator and denominator of the fraction by the same expression so that the denominator a power of the type of root. (Since you have square roots we will be making the denominator a perfect square. If they had been cube roots, we would make the denominator a perfect cube. If they had been 4th roots, etc.)</li><li>Use the {{{root(a, p)/root(a, q) = root(a, p/q)}}} property again, this time from right to left, to rewrite the radical of a fraction as a fraction of radicals.</li><li>Simplify the two radicals. If step 3 was done correctly then the radical in the denominator should disappear.</li></ol>
Let's see this in action.
1) Use the property to combine the radicals:
{{{sqrt(x^3/12)}}}
2) Reduce the fraction inside the radical, if possible.
This fraction, {{{x^3/12}}}, will not reduce.
3) Multiply the numerator and denominator so that the denominator inside the radical becomes a power of the type of root.
In this expression we will make the denominator into a perfect square. The obvious choice would be to multiply the numerator and denominator by 12. But we can do better: Multiplying the numerator and denominator by 3. This will make the denominator a 36 which is a perfect square. The lower perfect square saves us from extra simplifying later on.
{{{sqrt((x^3/12)(3/3))}}}
giving us:
{{{sqrt(3x^3/36)}}}
4) Use the property to split the radicals:
{{{sqrt(3x^3)/sqrt(36)}}}
5) Simplify the radicals.
The denominator, being the square root of a perfect square, simplifies to a nice whole number. The numerator is not a perfect square. But it does have a perfect square factor:
{{{sqrt(3x^2*x)/6}}}
{{{sqrt(x^2*3x)/6}}}
{{{(sqrt(x^2)*sqrt(3x))/6}}}
{{{(x*sqrt(3x))/6}}}
This is the simplified answer.<br>
When we were making the denominator a perfect square we turned it into a 36. If we had used the more obvious choice, here is how it would have worked out:
{{{sqrt((x^3/12)(12/12))}}}
{{{sqrt((12x^3)/144)}}}
{{{sqrt(12x^3)/sqrt(144)}}}
{{{sqrt(4*3*x^2*x)/12}}}
{{{sqrt(4*x^2*3*x)/12}}}
{{{(sqrt(4)*sqrt(x^2)*sqrt(3*x))/12}}}
{{{(2*x*sqrt(3*x))/12}}}
{{{(2*x*sqrt(3*x))/(2*6)}}}
{{{(cross(2)*x*sqrt(3*x))/(cross(2)*6)}}}
{{{(x*sqrt(3*x))/6}}}
We get the same answer. But did you notice the extra simplifying?