Question 390985
{{{5sqrt(-1/243)}}}
With the negative radicand of a square root, this will be an imaginary number. So the first thing we do is handle the minus. Start by factoring out -1:
{{{5sqrt(-1*(1/243))}}}
Next we use a property of radicals, {{{root(a, p*q) = root(a, p)*root(a, q)}}}, to separate the factors into their own square roots:
{{{5sqrt(-1)*sqrt(1/243)}}}
Since {{{sqrt(-1)}}} = i this becomes:
{{{5i*sqrt(1/243)}}}
Next we simplify. First we can use another property of radicals, {{{root(a, p/q) = root(a, p)/root(a, q)}}}, to separate the numerator and denominator into their own square roots:
{{{5i*(sqrt(1)/sqrt(243))}}}
which simplifies as follows:
{{{5i*(1/sqrt(81*3))}}}
{{{5i*(1/(sqrt(81)*sqrt(3)))}}}
{{{5i*(1/(9*sqrt(3)))}}}
To rationalize the denominator we just multiply the numerator and denominator by {{{sqrt(3)}}}:
{{{5i*(1/(9*sqrt(3)))(sqrt(3)/sqrt(3))}}}
{{{5i*(sqrt(3)/(9*sqrt(9)))}}}
{{{5i*(sqrt(3)/(9*3))}}}
{{{5i*(sqrt(3)/27)}}}
This is a simplified answer. However imaginary numbers are usually written in the form of some number times "i". Rearranging the above into this form we get:
{{{(5sqrt(3)/27)i}}}