Question 275568
{{{sqrt( 12a^2b)* sqrt(15ab^2)}}}
One of the properties of radicals is: {{{root(a, p)*root(a, q) = root(a, p*q)}}}
We will ue this property in both "directions". We will start by using it to multiply the two square roots into one:
{{{sqrt( 12a^2b*15ab^2)}}}
Simplifying the radicand (the expression inside a radical) we get:
{{{sqrt( 180a^3b^3)}}}
The last stage is to simplify the square root. This involves finding factors of the radicand that are perfect squares:
{{{sqrt( 36*5*a^2*a*b^2*b)}}}
Now we will use the property in the other direction to separate the factors into their own square roots:
{{{sqrt( 36)*sqrt(5)*sqrt(a^2)*sqrt(a)*sqrt(b^2)*sqrt(b)}}}
Simplifying the square roots of the perfect squares we get:
{{{6*sqrt(5)*a*sqrt(a)*b*sqrt(b)}}}
And finally we'll use the Commutative Property of Multiplication to rearrange the factors and our property of radcials to re-combine all the remaining square roots:
{{{6ab*sqrt(5ab)}}}