How do I simplify $(\frac{x \sqrt{x} - y \sqrt{y}}{x - \sqrt{x} \sqrt{y}} - \frac{x - y}{\sqrt{x} + \sqrt{y}}) \cdot {(\frac{\sqrt{x} \sqrt{y} + 2 x}{3 x})}^{- 1}$ $((xsqrt(x)-ysqrt(y))/(x-sqrt(x)sqrt(y))-(x-y)/(sqrt(x)+sqrt(y)))*((sqrt(x)sqrt(y)+2x)/(3x))^-1$ ?

Question

How do I simplify $(\frac{x \sqrt{x} - y \sqrt{y}}{x - \sqrt{x} \sqrt{y}} - \frac{x - y}{\sqrt{x} + \sqrt{y}}) \cdot {(\frac{\sqrt{x} \sqrt{y} + 2 x}{3 x})}^{- 1}$ $((xsqrt(x)-ysqrt(y))/(x-sqrt(x)sqrt(y))-(x-y)/(sqrt(x)+sqrt(y)))*((sqrt(x)sqrt(y)+2x)/(3x))^-1$ ?

1 Answer

Impact of this question

2689 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-04-22T16:02:19

$3 \sqrt{y}$ $3 sqrt y$

Explanation:

Ugh, that's a messy one. Let's try to control the mess by writing $a = \sqrt{x}$ $a=\sqrt{x}$ so $x = a^{2}$ $x=a^2$ and $b = \sqrt{y}$ $b=sqrt{y}$ so $y = b^{2}$ $y=b^2$ .

$(\frac{x \sqrt{x} - y \sqrt{y}}{x - \sqrt{x} \sqrt{y}} - \frac{x - y}{\sqrt{x} + \sqrt{y}}) {(\frac{\sqrt{x} \sqrt{y} + 2 x}{3 x})}^{- 1}$ $( frac{ x sqrt{x} - y sqrt{y} }{x - sqrt{x} sqrt{y} } - frac { x- y} {sqrt x + sqrt y} )( frac{sqrt{x} sqrt{y} + 2x }{3x})^{-1}$

= $(\frac{a^{3} - b^{3}}{a^{2} - a b} - \frac{a^{2} - b^{2}}{a + b}) {(\frac{a b + 2 a^{2}}{3 a^{2}})}^{- 1}$ $( frac{ a^3 - b^3}{a^2 -ab } - frac { a^2-b^2} {a+b} )( frac{a b + 2a^2 }{3a^2})^{-1}$

Well that's less messy already, isn't it? It's easier to see the factors this way:

= $(\frac{(a - b) (a^{2} + a b + b^{2})}{a (a - b)} - \frac{(a + b) (a - b)}{a + b}) (\frac{3 a^{2}}{a (2 a + b)})$ $( frac{ (a-b)(a^2 + ab + b^2) }{a(a-b)} - frac { (a+b)(a-b)} {a+b} )( frac{3a^2}{a( 2a + b) })$

We got some cancelling we can do,

$= (\frac{a^{2} + a b + b^{2}}{a} - (a - b)) (\frac{3 a}{2 a + b})$ $= ( frac{ a^2 + ab + b^2 }{a} - (a-b) )( frac{3a}{ 2a + b })$

$= (\frac{a^{2} + a b + b^{2} - a^{2} + a b}{a}) (\frac{3 a}{2 a + b})$ $= ( frac{ a^2 + ab + b^2 - a^2 + ab}{a} )( frac{3a}{ 2a + b })$

$= (\frac{b (2 a + b)}{a}) (\frac{3 a}{2 a + b})$ $= ( frac{ b(2a + b) }{a} )( frac{3a}{ 2a + b })$

$= 3 b$ $= 3 b$

$= 3 \sqrt{y}$ $= 3 sqrt y$

Science

Math

Humanities

... and beyond

How do I simplify $(\frac{x \sqrt{x} - y \sqrt{y}}{x - \sqrt{x} \sqrt{y}} - \frac{x - y}{\sqrt{x} + \sqrt{y}}) \cdot {(\frac{\sqrt{x} \sqrt{y} + 2 x}{3 x})}^{- 1}$ $((xsqrt(x)-ysqrt(y))/(x-sqrt(x)sqrt(y))-(x-y)/(sqrt(x)+sqrt(y)))*((sqrt(x)sqrt(y)+2x)/(3x))^-1$ ?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do I simplify (x√x−y√yx−√x√y−x−y√x+√y)⋅(√x√y+2x3x)−1((xsqrt(x)-ysqrt(y))/(x-sqrt(x)sqrt(y))-(x-y)/(sqrt(x)+sqrt(y)))*((sqrt(x)sqrt(y)+2x)/(3x))^-1 ?

1 Answer

Explanation:

Related questions

Impact of this question

How do I simplify $(\frac{x \sqrt{x} - y \sqrt{y}}{x - \sqrt{x} \sqrt{y}} - \frac{x - y}{\sqrt{x} + \sqrt{y}}) \cdot {(\frac{\sqrt{x} \sqrt{y} + 2 x}{3 x})}^{- 1}$ $((xsqrt(x)-ysqrt(y))/(x-sqrt(x)sqrt(y))-(x-y)/(sqrt(x)+sqrt(y)))*((sqrt(x)sqrt(y)+2x)/(3x))^-1$ ?