I saw a problem recently that looked like this:

Assume $w$ and $z$ are positive. If $z^{4w} = 64$, what does $z^{6w}$ equal?

And I had absolutely no idea how to even begin attempting this equation. How could I have gone about figuring out the answer? This question was multiple choice, though I don't remember the answers available to me.

  • $\begingroup$ What's the relationship between $a,b,z,w$? $\endgroup$ – Najib Idrissi Apr 28 '14 at 14:00
  • $\begingroup$ Sorry, "a" and "b" were mistakes on my part. It should have been "z" and "w" only. $\endgroup$ – Noah Crowley Apr 28 '14 at 14:07
  • $\begingroup$ Well then, do you know that $x^{ab} = (x^a)^b$? $\endgroup$ – Najib Idrissi Apr 28 '14 at 14:09
  • $\begingroup$ Yes, but I'm not sure how to apply that knowledge to this problem. $\endgroup$ – Noah Crowley Apr 28 '14 at 14:20
  • $\begingroup$ $4 \times \frac{3}{2} = 6$. $\endgroup$ – Najib Idrissi Apr 28 '14 at 14:21

Given $z^{4w} = 64, z^{6w} =$ ?

We need to find a relationship between the exponents. Since $x^{ab} = (x^{a})^b$,

$$\Large{z^{6w} = z^{4w\cdot1.5}} = (z^{4w})^{1.5} = 64^{1.5} = 512$$

  • $\begingroup$ Oh! Thank you! Now it makes sense!! :D $\endgroup$ – Noah Crowley Apr 28 '14 at 14:51

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