Questions: Question 12 1 pts Use rules of exponents to rewrite the following expression without any fractions (using negative exponents if needed). ((r^6 y^6 z^2 / x) / (r^5 x^3 y z^4))^(-5) ⋅ ((r^4 x^6 y^2 z^2) / (x y^3 z^6 / r^3))^(2) x^30 y^-27 z^2 r^9 x^-30 y^27 z^-2 r^-9 x^-30 y^-27 z^-2 r^-9 x^30 y^27 z^2 r^9 None of these are correct.

$Question 12 1 pts Use rules of exponents to rewrite the following expression without any fractions (using negative exponents if needed). ((r^6 y^6 z^2 / x) / (r^5 x^3 y z^4))^(-5) ⋅ ((r^4 x^6 y^2 z^2) / (x y^3 z^6 / r^3))^(2) x^30 y^-27 z^2 r^9 x^-30 y^27 z^-2 r^-9 x^-30 y^-27 z^-2 r^-9 x^30 y^27 z^2 r^9 None of these are correct.$

Transcript text: Question 12 1 pts Use rules of exponents to rewrite the following expression without any fractions (using negative exponents if needed). \[ \left(\frac{\frac{r^{6} y^{6} z^{2}}{x}}{r^{5} x^{3} y z^{4}}\right)^{-5} \cdot\left(\frac{r^{4} x^{6} y^{2} z^{2}}{\frac{x y^{3} z^{6}}{r^{3}}}\right)^{2} \] $x^{30} y^{-27} z^{2} r^{9}$ $x^{-30} y^{27} z^{-2} r^{-9}$ $x^{-30} y^{-27} z^{-2} r^{-9}$ $x^{30} y^{27} z^{2} r^{9}$ None of these are correct.

Solution

Solution Steps

To solve this problem, we need to simplify the given expression using the rules of exponents. We will first simplify each fraction inside the parentheses, then apply the exponent outside the parentheses, and finally combine the results.

Step 1: Simplify the Inner Fractions

First, we simplify the inner fractions of the given expression: \[ \left(\frac{\frac{r^{6} y^{6} z^{2}}{x}}{r^{5} x^{3} y z^{4}}\right)^{-5} \cdot \left(\frac{r^{4} x^{6} y^{2} z^{2}}{\frac{x y^{3} z^{6}}{r^{3}}}\right)^{2} \]

For the first fraction: \[ \frac{\frac{r^{6} y^{6} z^{2}}{x}}{r^{5} x^{3} y z^{4}} = \frac{r^{6} y^{6} z^{2}}{x} \cdot \frac{1}{r^{5} x^{3} y z^{4}} = \frac{r^{6-5} y^{6-1} z^{2-4}}{x^{1+3}} = \frac{r y^{5} z^{-2}}{x^{4}} \]

For the second fraction: \[ \frac{r^{4} x^{6} y^{2} z^{2}}{\frac{x y^{3} z^{6}}{r^{3}}} = \frac{r^{4} x^{6} y^{2} z^{2}}{x y^{3} z^{6}} \cdot r^{3} = \frac{r^{4+3} x^{6-1} y^{2-3} z^{2-6}}{1} = r^{7} x^{5} y^{-1} z^{-4} \]

Step 2: Apply the Outer Exponents

Next, we apply the outer exponents to the simplified fractions: \[ \left(\frac{r y^{5} z^{-2}}{x^{4}}\right)^{-5} = \left(r y^{5} z^{-2} x^{-4}\right)^{-5} = r^{-5} y^{-25} z^{10} x^{20} \]

\[ \left(r^{7} x^{5} y^{-1} z^{-4}\right)^{2} = r^{14} x^{10} y^{-2} z^{-8} \]

Step 3: Combine the Results

Finally, we combine the results of the two expressions: \[ r^{-5} y^{-25} z^{10} x^{20} \cdot r^{14} x^{10} y^{-2} z^{-8} = r^{-5+14} x^{20+10} y^{-25-2} z^{10-8} = r^{9} x^{30} y^{-27} z^{2} \]