Question 201791
To simplify {{{sqrt (56ab^3) / sqrt (7a)}}} we will need a couple of basic properties of square roots:
{{{sqrt(x/y) = sqrt(x)/sqrt(y)}}} and
{{{sqrt(x*y) = sqrt(x)*sqrt(y)}}}
We can use the first property to combine your fraction of square roots into a square root of a fraction:
{{{sqrt (56ab^3) / sqrt (7a) = sqrt((56ab^3) / (7a))}}}
We do this because we can cancel factors and reduce the fraction:
{{{sqrt((56ab^3) / (7a)) = sqrt((7*8*a*b^3)/(7a))}}}
The 7's and the a's cancel leaving:
{{{sqrt(8b^3)}}}
Now we factor out as many perfect squares as we can find:
{{{sqrt(4*2*b^2*b)}}}
We can use the Commutative property to rearrange the order:
{{{sqrt(4*b^2*2b)}}}
And now we can use the second property to separate out the perfect square factors:
{{{sqrt(4)*sqrt(b^2)*sqrt(2b)}}}
The {{{sqrt(4) = 2}}}.
And normally we would say that {{{sqrt(b^2) = abs(b)}}} so that we can guarantee a non-negative value for the value of the square root. But if we look back at the original expression, we can see that:<ol><li>"a" cannot be zero because it would make the denominator zero.</li><li>"a" cannot be negative because the radicand (the number inside the square root) must not be negative</li><li>So "a" must be positive.</li><li>Since "a" is positive {{{b^3}}} must be zero or positive so that the radicand in the numerator is not negative.</li><li>Since {{{b^3}}} is zero or positive then "b" must be zero or positive.</li><li>If "b" is zero or positive, then {{{abs(b) = b}}}</li></ol>
So we do not need |b| to guarantee a non-negative square root. We can use just plain "b".
Substituting into our expression we get:
{{{2*b*sqrt(2b)}}}
or simply
{{{2b*sqrt(2b)}}}