Question

what expression is equivalent to √x^-3 y^-7 z^2 if x, y, and z are positive real numbers

Answer 1

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:

\[
\frac{z}{x^{3/2} y^{7/2}}
\]