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$$\frac{(z x^{-1} y)^5 y^5}{x^{-4} z^{-4}}$$

How would I evaluate this if $x = 10$, $y = -3$ and $z = 3$ ? I would like a step-by-step solution to help me fully understand it.

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up vote 5 down vote accepted

You can simply plug in and chug.

  1. Multiply $z=3$ by $x^{-1} = \frac{1}{10}$; get $A$.
  2. Multiply $A$ by $y=-3$ to get $B$. This is $(zx^{-1}y)$.
  3. Raise $B$ to the fifth power (multiply it by itself five times) to get $C$; that is, $C=(zx^{-1}y)^5$.
  4. Raise $y=-3$ to the fifth power to get $D$; this is $y^5$.
  5. Multiply $C$ and $D$ together to get $E$. This is the numerator of your expression.
  6. Take the reciprocal of $x^4$ (that is, $10^4$, then one over what you get). Call that $F$.
  7. Take the reciprocal of $z^4$; call that $G$.
  8. Multiply $F$ by $G$ and call it $H$; this is $x^{-4}z^{-4}$.
  9. Divide $E$ by $H$. That's the final answer.

Or you can do some algebra first: dividing by $x^{-4}$ is the same as multiplying by $x^4$. Same with $z^{-4}$. So your expression is exactly the same as $$ x^4z^4(zx^{-1}y)^5y^5.$$ And since $(ab)^n = a^nb^n$, then $(zx^{-1}y)^5 = z^5 (x^{-1})^5 y^5 = z^5 x^{-5}y^5$. So your expression is the same as $$x^4 z^4 z^5 x^{-5}y^5y^5.$$ Now simplify by putting all the $x$s, all the $y$s and all the $z$s together, plug in the values you have, and perform the operations that are left.

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Thanks! Helped a lot, I was just having problems with procedure. – user7759 Mar 4 '11 at 0:07

First, use the fact that $x^{-n}=\frac{1}{x^n}$ to find $$\frac{(z x^{-1} y)^5 y^5}{x^{-4} z^{-4}}=(zx^{-1}y)^5 \cdot \frac{y^5}{x^{-4}z^{-4}}=\left(\frac{zy}{x}\right)^5\cdot x^4z^4y^5 $$ Next, $$\left(\frac{zy}{x}\right)^5\cdot x^4z^4y^5=\frac{z^5y^5}{x^5}\cdot x^4z^4y^5=z^9y^{10}x$$ Can you finish from here?

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