To simplify the expression \(\sqrt{x^{-3} y^{-7} z^{2}}\), we can rewrite the square root in terms of the exponent:
\[ \sqrt{x^{-3} y^{-7} z^{2}} = (x^{-3} y^{-7} z^{2})^{1/2} \]
Now, we can apply the exponent to each factor separately:
\[ = x^{-3/2} y^{-7/2} z^{2/2} \]
This simplifies to:
\[ = x^{-3/2} y^{-7/2} z^{1} \]
To write it in a more standard form, we can express the negative exponents as fractions:
\[ = \frac{z}{x^{3/2} y^{7/2}} \]
Thus, the expression equivalent to \(\sqrt{x^{-3} y^{-7} z^{2}}\) is:
\[ \frac{z}{x^{3/2} y^{7/2}} \]
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