# Variable Base with Variable as Factor in Exponent, Find Value

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.

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

Given $z^{4w} = 64, z^{6w} =$ ?
We need to find a relationship between the exponents. Since $x^{ab} = (x^{a})^b$,
$$\Large{z^{6w} = z^{4w\cdot1.5}} = (z^{4w})^{1.5} = 64^{1.5} = 512$$