Can someone walk me through how this expression simplifies to y/x?

Question

I am just wondering how this equation comes to be: it is from an economics problem involving marginal utilities. I have my two variables, $x$ and $y$.

Intuitively, how does $$\frac{0.5\times x^{-0.5}\times y^{0.5}}{0.5\times x^{0.5}\times y^{-0.5}}= \frac{y}{x}?$$

I'm guessing that the $0.5$ and $(1/2)$ both cancel out, but I am not sure what happens next.

Thank you!

Answer 1

Hint: $\frac{x^{-0.5}}{x^{0.5}}=x^{-0.5-0.5}=x^{-1}=\frac{1}{x}$ and $\frac{y^{0.5}}{y^{-0.5}}=y^{0.5-(-0.5)}=y^{0.5+0.5}=y^1=y$

Answer 2

Hint: $$x^{-0.5}=\frac{1}{x^{0.5}}=\frac{1}{\sqrt{x}}$$ and similarly $$\frac{1}{y^{-0.5}}=\frac{1}{\frac{1}{y^{0.5}}}=y^{0.5}=\sqrt{y\phantom{!}}$$ Now $$x^{0.5}\times x^{0.5}=x^{0.5+0.5}=x^1=x$$

Answer 3

$x^{0.5}=\sqrt{x}$ and $x^{-0.5}=\dfrac{1}{\sqrt{x}}$

So $(0.5x^{-0.5} \times y^{0.5})/(0.5x^{0.5} \times y^{-0.5}) =\dfrac{\frac12 \times \frac{1}{\sqrt{x}}\times {\sqrt{y}}}{\frac12 \times {\sqrt{x}}\times \frac{1}{\sqrt{y}}}$.

Multiply numerator and denominator by $2 \times {\sqrt{x}}\times \sqrt{y}$ to get the result.

Answer 4

Let me show you. It's just the "Multiply by one" trick:

$\frac{0.5 x^{-\frac{1}{2}} y^\frac{1}{2}}{0.5 x^\frac{1}{2} y^{-\frac{1}{2}}} \cdot 1 \cdot 1 = \frac{0.5 x^{-\frac{1}{2}} y^\frac{1}{2}}{0.5 x^\frac{1}{2} y^{-\frac{1}{2}}} \cdot \frac{y^\frac{1}{2}}{y^\frac{1}{2}} \cdot \frac{x^\frac{1}{2}}{x^\frac{1}{2}}$

You know, of course, that $a^p a^q = a^{p+q}$:

$\frac{0.5 x^{-\frac{1}{2}+\frac{1}{2}} y^{\frac{1}{2}+\frac{1}{2}}}{0.5 x^{\frac{1}{2}+\frac{1}{2}} y^{-\frac{1}{2}+\frac{1}{2}}}=\frac{0.5x^0 y^1}{0.5x^1 y^0} = \frac{y}{x}$

And there you have it :)