Question

Rationalize the denominator and simplify: (sqrt= square root)

2 sqrt X - 3 sqrt Y
-------------------------
sqrt X + sqrt Y

Answer 1

Explanation:

`Domain:\ x>0\ \wedge\ y>0`

`(2√(x)-3√(y))/(√(x)+√(y))\qquad|\text{use}\ (a+b)(a-b)=a^2-b^2\\\\=(2√(x)-3√(y))/(√(x)+√(y))\cdot(√(x)-√(y))/(√(x)-√(y))=((2√(x)-3√(y))(√(x)-√(y)))/((√(x)+√(y))(√(x)-√(y)))\\\\\text{use FOIL}\ (a+b)(c+d)=ac+ad+bc+bd\\\\=((2√(x))(√(x))+(2√(x))(-√(y))+(-3√(y))(√(x))+(-3√(y))(-√(y)))/((√(x))^2-(√(y))^2)`

`\text{use}\ (√(a))^2=a\ \text{and}\ (√(a))(√(a))=a\ \text{for}\ a\geq0\\\text{and}\ √(ab)=√(a)\cdot√(b)\ \text{for}\ a\geq0\ \wedge\ b\geq0\\\\=(2x-2√(xy)-3√(xy)+3y)/(x-y)\qquad|\text{combine like terms}\\\\=(2x+3y-5√(xy))/(x-y)`