```Question 216485
Simplifying square roots depends on understanding how to use the properties of square roots:<ul><li>{{{sqrt(a*b) = sqrt(a)*sqrt(b)}}}</li><li>{{{sqrt(a/b) = sqrt(a)/sqrt(b)}}}</li></ul>
Since we have no fractions in this problem we will not need the second property. The first property allows us to take the square root of a product (multiplication) and rewrite it as a product of square roots. And, as we will see, finding perfect square factors (not including 1) and using this property will allow us to simplify square roots.<br>
So we start to simplify {{{sqrt(675)}}} by trying to find perfect square factors (except 1) of 675. Perfect squares are numbers resulting from squaring a number (1, 4, 9, etc.). Ideally we want to find the largest perfect square factor but since we don't know the multiplication facts for all perfect squares we may not find the largest one right away.<br>
Perhaps you notice that the digits of 675 add up to 18 which is divisible by 9. This means 675 is also divisible by 9 (which is a perfect square). Or perhaps you notice that 675 will be divisible by 25 (which is another perfect square). We can start with either one (or both). We'll use 25:
{{{sqrt(675) = sqrt(25*27) = sqrt(25)*sqrt(27)}}}
Since 25 may not be the largest perfect square factor we need to see if 27 has a perfect square factor. And it does, 9:
{{{sqrt(675) = sqrt(25*27) = sqrt(25)*sqrt(27) = sqrt(25)*sqrt(9*3) = sqrt(25)*sqrt(9)*sqrt(3)}}}
Since 3 has no perfect square factors (other than 1 which we don't use), we are finished with finding perfect square factors. Looking at {{{sqrt(25)*sqrt(9)*sqrt(3)}}} we should be able to see why we found the perfect square factors. {{{sqrt(25) = 5}}} and {{{sqrt(9) = 3}}}. Substituting these values into our expression we get:
{{{sqrt(675) = sqrt(25*27) = sqrt(25)*sqrt(27) = sqrt(25)*sqrt(9)*sqrt(3) = 5*3*sqrt(3) = 15sqrt(3)}}}
So {{{sqrt(675) = 15sqrt(3)}}}.```