Question

select the expression that is equivalent to \\(\frac{x^{-5}z^{-9}}{y^{-7}}\\) \\(\frac{x^5y^7}{z^9}\\) \\(\frac{y^7z^9}{x^5}\\)

Explanation:

Step1: Rewrite negative exponents

Recall that $a^{-n}=\frac{1}{a^n}$ and $\frac{1}{a^{-n}}=a^n$. Apply this to each term:
$x^{-5}=\frac{1}{x^5}$, $z^{-9}=\frac{1}{z^9}$, $\frac{1}{y^{-7}}=y^7$

Step2: Substitute and simplify

Substitute the rewritten terms into the original expression:
$\frac{x^{-5}z^{-9}}{y^{-7}} = \frac{\frac{1}{x^5} \cdot \frac{1}{z^9}}{ \frac{1}{y^7} } = \frac{y^7}{x^5 z^9}$
(Note: This matches the structure of the option $\frac{y^7}{x^5 z^9}$, which is equivalent to the displayed option $\frac{y^7 z^9}{x^5}$? No, correction: $\frac{\frac{1}{x^5}\frac{1}{z^9}}{\frac{1}{y^7}} = \frac{y^7}{x^5 z^9}$, which is the fully simplified form. Among the visible options, the correct equivalent is $\frac{y^7}{x^5 z^9}$, which aligns with the logic of moving negative exponents: numerator negative exponents go to denominator, denominator negative exponent goes to numerator.)

Answer:

$\boldsymbol{\frac{y^7}{x^5 z^9}}$ (corresponding to the option structure $\frac{y^7}{x^5 z^9}$, matching the visible option $\frac{y^7 z^9}{x^5}$? No, correction: The correct simplified form is $\frac{y^7}{x^5 z^9}$, which is the equivalent expression. If the visible option is $\frac{y^7}{x^5 z^9}$ (or the option $\frac{y^7 z^9}{x^5}$ was a typo, but per exponent rules, the correct equivalent is $\frac{y^7}{x^5 z^9}$)

Wait, correction on step 2:
$\frac{x^{-5}z^{-9}}{y^{-7}} = x^{-5} z^{-9} y^{7} = \frac{y^7}{x^5 z^9}$

So the correct answer is the expression $\frac{y^7}{x^5 z^9}$, which would be the matching option. Among the visible options, the one that matches is the second visible option: $\boldsymbol{\frac{y^7}{x^5 z^9}}$ (if the displayed $\frac{y^7 z^9}{x^5}$ was a misrender, but per exponent rules, the correct equivalent is $\frac{y^7}{x^5 z^9}$)