Question 398495
{{{sqrt(450)/sqrt(1800)}}}
With expression like this, a square root over a square root, I like to use the following procedure:<ol><li>Use the property {{{root(a, p)/root(a, q) = root(a, p/q)}}} to combine the two square roots into a single square root.</li><li>Reduce the fraction inside the square root, if possible.</li><li>If the denominator is not a perfect square, then multiply the numerator and denominator by some number so that the denominator becomes a perfect square.</li><li>Use the same property as step 1, only in reverse, to split the square root into a square root over a square root.</li><li>Simplify the square roots.</li></ol>
Let's see this in action.
1) Combine the square roots:
{{{sqrt(450/1800)}}}
2) Reduce the fraction inside. Since 450*4 = 1800 this fraction reduces to:
{{{sqrt(1/4)}}}
3) If the denominator is is not a perfect square, then make it one. Since 4 is a perfect square, we can skip this step.
4) Split the square root:
{{{sqrt(1)/sqrt(4)}}}
Since both 1 and 4 are perfect squares, both the numerator and denominator simplify to nice whole numbers:
{{{1/2}}}<br>
Note 1: The procedure I've described is not the only way to simplify your expression. But it is pretty efficient at handing square roots over square roots.
Note 2: Any procedure used correctly should simplify your expression to a 1/2.