Question 352886
{{{3sqrt(-1/125)}}}
With a negative radicand (expression with a radical), this expression is going to be an imaginary number. And with imaginary numbers we use i to factor out the factor of -1:
{{{3sqrt(-1*(1/125))}}}
{{{3sqrt(-1)sqrt(1/125)}}}
{{{3i*sqrt(1/125)}}}
Next, a simplified radical expression has ...
- No radicals in a denominator
- No fractions in a radicand
We have a fraction in the radicand so we have more work to do. Although there are other ways to simplify radicands with fractions, I like to start by making the denominator a perfect square. So we'll start by figuring out what we can multiply 125 by to turn it into a perfect square. Surely multiplying by 125 will work. But there is a smaller number that will work. And since smaller numbers are easier to work with, I prefer to go that way. Since {{{125 = 5*5^2}}} I can see that just another factor of 5 will make 125 a perfect square, It will be the perfect square of 5*5 or 25:
{{{3i*sqrt((1/125)(5/5))}}}
{{{3i*sqrt(5/625)}}}
Now we'll split the fraction into spearate square roots:
{{{3i*sqrt(5)/sqrt(625)}}}
and replace {{{sqrt(625)}}} with 25:
{{{3i*sqrt(5)/25}}}
This is our simplified expression. An alternate form, which may be preferred by your teacher, would be:
{{{(3sqrt(5)/25)i}}}