<PRE><B>Question 398823</B>
With expressions like these, a radical over the same kind of radical, I like to use the following procedure:&lt;ol&gt;&lt;li&gt;Use the property {{{root(a, p)/root(a, q) = root(a, p/q)}}} to combine the two radicals into a single radical.&lt;/li&gt;&lt;li&gt;Reduce the fraction inside the radical, if possible.&lt;/li&gt;&lt;li&gt;If the denominator is not a perfect power of the type of radical, then multiply the numerator and denominator by some expression so that the denominator becomes a perfect power of the type of radical.&lt;/li&gt;&lt;li&gt;Use the same property as step 1, only in reverse, to split the radical back to a radical over a radical.&lt;/li&gt;&lt;li&gt;Simplify. If the previous steps were done correctly, there should be no radicals remaining in the denominator.&lt;/li&gt;&lt;/ol&gt;
Let&#39;s see this in action.
{{{root(3, 170x)/root(3, 10xy^2)}}}
1) Combine radicals:
{{{root(3, 170x/10xy^2)}}}
2) Reduce the fraction inside. The x&#39;s cancel and a factor of 10 cancels leaving:
{{{root(3, 17/y^2)}}}
3) If the denominator is is not a perfect power of the type of radcial, then multiply the numerator and denominator by whatever makes the denominator a perfect power of the type of radical. Since we are working with cube roots, we are looking to make the denominator a perfect cube. All we need is another factor of y:
{{{root(3, (17/y^2)(y/y))}}}
which simplifies to:
{{{root(3, 17y/y^3)}}}
4) Split the radical:
{{{root(3, 17y)/root(3, y^3)}}}
5. Simplify.
{{{root(3, 17y)/y}}}
Since there are no perfect cube factors in 17y, the radical in the numerator will not simplify further.&lt;br&gt;
{{{root(3, x^2)/root(3 4)}}}
1. Combine the radicals:
{{{root(3, x^2/4)}}}
2. Reduce the fraction. This fraction will not reduce.
3. Make the denominator a power of the type of radical. The &quot;nearest&quot; perfect cube to 4 is 8. So we just have to multiply the numerator and denominator by 2:
{{{root(3, (x^2/4)(2/2))}}}
which simplifies to:
{{{root(3, 2x^2/8)}}}
4. Split the radical:
{{{root(3, 2x^2)/root(3, 8)}}}
5. Simplify.
{{{root(3, 2x^2)/2}}}
There are no perfect cube factors in {{{2x^2}}} so the numerator will no simplify any further.&lt;br&gt;
Note 1: The procedure I&#39;ve described is not the only way to simplify your expressions. But it is pretty efficient at handing radical over radical expression.
Note 2: Any procedure used correctly should simplify your expressions to the same results as we got above.
Note 3: If the radicals are of different types then use fractional exponents to make them the same type. For example:
{{{sqrt(2x)/root(3, y^2)}}}
{{{(2x)^(1/2)/(y^2)^(1/3)}}}
{{{(2x)^(3/6)/(y^2)^(2/6)}}}
{{{((2x)^3)^(1/6)/((y^2)^2)^(1/6)}}}
{{{root(6, (2x)^3)/root(6, (y^2)^2)}}}
{{{root(6, 8x^3)/root(6, y^4)}}}
Now that the radicals are both 6th roots, we can use the procedure above:
1. Combine:
{{{root(6, 8x^3/y^4)}}}
2. Reduce. This will not reduce.
3. Make the denominator a power of the type of radical:
{{{root(6, (8x^3/y^4)(y^2/y^2))}}}
{{{root(6, (8x^3y^2)/y^6)}}}
4. Split
{{{root(6, 8x^3y^2)/root(6, y^6)}}}
5. Simplify. Since sixth roots are supposed to be positive and since we do not know if y is positive, we should use absolute value when simplifying {{{root(6, y^6)}}}:
{{{root(6, 8x^3y^2)/abs(y)}}}</PRE>