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I got this equation

$$MRS= - \frac{3/4x^{1/4} \ y^{-1/2}}{\frac{}{}3/2x^{-1/4}\ y^{1/2}}$$

As you can see the absolute values of the exponents are the same so they should cancel out each other somehow, but I don't know the rule.

Can somebody explain and simplify the equation?

Teyzer18
  • 545
Ivan Hanák
  • 101

2 Answers2

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The rule is that $$\frac{x^a}{x^b}=x^{a-b}.$$ You can gain intuition for this by thinking about the case when $a$ and $b$ are positive integers with $a>b$: the numerator represents $x$ multiplied together $a$ times, and the denominator is $x$ multiplied together $b$ times. If you take divide these two, you should get $a-b$ copies of $x$.

In particular, this tells you that $$\frac{x^a}{x^{-a}}=x^{2a}.$$ Can you use this to simplify your expression?

Carl Schildkraut
  • 32,931
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Using $x^{-n} = \frac{1}{x^n}$, we can rewrite the equation as

$$MRS = \frac{3 \times 2}{3 \times 4} \times x^{1/4}x^{-(-1/4)}y^{-1/2}y^{-1/2}$$

$$ = \frac{1}{2} \times x^{1/4 + 1 /4}y^{-1/2-1/2} = \frac{1}{2}x^{2/4}y^{-1}$$

Using the fact that $\sqrt{x} = x^{1/2}$, we can write

$$MRS = \frac{\sqrt{x}}{2y}$$

Teyzer18
  • 545