Question 624840
{{{((5x)((3y^2)/2)^(1/2))-((3y)((8x^2)/3)^(1/2))+((2)((3x^2y^2)/2)^(1/2))}}}
Since exponents of 1/2 mean the same thing as square roots, I am going to rewrite the expression with square roots:
{{{(5x)sqrt((3y^2)/2)-(3y)sqrt((8x^2)/3)+(2)sqrt((3x^2y^2)/2)}}}
Next we will simplify each square root. Part of simplifying square roots is rationalizing the denominators. There are a variety of ways to go about doing this. I like to start by making each denominator a perfect square:
{{{(5x)sqrt(((3y^2)/2)(2/2))-(3y)sqrt(((8x^2)/3)(3/3))+(2)sqrt(((3x^2y^2)/2)(2/2))}}}
which leads to:
{{{(5x)sqrt((6y^2)/4)-(3y)sqrt((24x^2)/9)+(2)sqrt((6x^2y^2)/4)}}}
Next we use the {{{root(a, p/q) = root(a, p)/root(a, q)}}} property of radicals to split each square root:
{{{(5x)sqrt(6y^2)/sqrt(4)-(3y)sqrt(24x^2)/sqrt(9)+(2)sqrt(6x^2y^2)/sqrt(4)}}}
Because of our earlier work, each denominator will simplify:
{{{(5x)sqrt(6y^2)/2-(3y)sqrt(24x^2)/3+(2)sqrt(6x^2y^2)/2}}}
Next we simplify the square roots in the numerators. Each one happens to have one or more prefect square factors (which I like to put first using the Commutative Property of Multiplication):
{{{(5x)sqrt(y^2*6)/2-(3y)sqrt(4*x^2*6)/3+(2)sqrt(x^2*y^2*6)/2}}}
Now we use another property of radicals, {{{root(a, p*q) = root(a, p)*root(a, q)}}}, to split the square roots so that each perfect square factor is in its own square root:
{{{(5x)(sqrt(y^2)*sqrt(6))/2-(3y)(sqrt(4)*sqrt(x^2)*sqrt(6))/3+(2)(sqrt(x^2)*sqrt(y^2)*sqrt(6))/2}}}
Each of the square roots of the perfect squares simplify:
{{{(5x)(y*sqrt(6))/2-(3y)(2*x*sqrt(6))/3+(2)(x*y*sqrt(6))/2}}}
which simplifies further to:
{{{(5xy*sqrt(6))/2-(6xy*sqrt(6))/3+(2xy*sqrt(6))/2}}}
The last two fractions reduce:
{{{(5xy*sqrt(6))/2-(2xy*sqrt(6))+(xy*sqrt(6))}}}
And last of all, these are all like terms! They are all {{{xy*sqrt(6)}}} terms. So we can add/subtract them. Just add/subtract the coefficients. To see this more easily, I'm going to rewrite the first and third terms so we can see the coefficient we should add:
{{{(5/2)xy*sqrt(6)-(2xy*sqrt(6))+(1xy*sqrt(6))}}}
which simplifies to:
{{{(3/2)xy*sqrt(6)}}}
since {{{(5/2)-2+1 = 3/2}}}
So
{{{((5x)((3y^2)/2)^(1/2))-((3y)((8x^2)/3)^(1/2))+((2)((3x^2y^2)/2)^(1/2))}}}
simplifies to
{{{(3/2)xy*sqrt(6)}}}
or, if you prefer the 1/2 exponents:
{{{(3/2)xy*(6)^(1/2)}}}