2050ba…: “Down the cloudy square”

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}[0]{1}$

Here is a walk-through example of how to do a problem like this:

Use rules of exponents to rewrite the following expression without any fractions (using negative exponents if needed):

$\left (\frac {\frac {a^3b^2c^{-4}d^6}{a^2d^{-2}}}{\frac {a^4c^{-2}}{c^3d}}\right )^5 \left (\frac {\frac {b^2cd^3}{a^2}}{\frac {b^3c}{a^{-2}d^3}}\right )^{-3}$ Solution: First, we should try to simplify the parts inside the large parentheses. Keep in mind this means we will need to do a bunch of algebra and our last line is likely to involve a ton of cancellation, so remember to use all the rules to combine exponentials with the same base to see how the below happens (especially the last step!)

$\left (\frac {\frac {a^3b^2c^{-4}d^6}{a^2d^{-2}}}{\frac {a^4c^{-2}}{c^3d}}\right )^5 \left (\frac {\frac {b^2cd^3}{a^2}}{\frac {b^3c}{a^{-2}d^3}}\right )^{-3}$	$=\left (\frac {a^3b^2c^{-4}d^6}{a^2d^{-2}}\cdot \frac {c^3d}{a^4c^{-2}}\right )^5 \left (\frac {b^2cd^3}{a^2}\cdot \frac {a^{-2}d^3}{b^3c}\right )^{-3}$	Step 1: Invert the bottom fractions and multiply




	$= \left (\frac {a^3b^2c^{-4}d^6c^3d}{a^2d^{-2}a^4c^{-2}}\right )^5 \left (\frac {b^2cd^3a^{-2}d^3}{a^2b^3c}\right )^{-3}$	Step 2: Multiply straight across




	$= \left (\frac {b^2cd^9}{a^3}\right )^5 \left (\frac {d^6}{a^4b}\right )^{-3}$	Step 3: Combine like terms to simplify.

Next we want to distribute the large power to each of the terms inside the parentheses. We can do this only because everything inside is being multiplied! In essence, the inside is factored, so we can distribute the outer power.

$\left (\frac {\frac {a^3b^2c^{-4}d^6}{a^2d^{-2}}}{\frac {a^4c^{-2}}{c^3d}}\right )^5 \left (\frac {\frac {b^2cd^3}{a^2}}{\frac {b^3c}{a^{-2}d^3}}\right )^{-3}$	$= \left (\frac {b^2cd^9}{a^3}\right )^5 \left (\frac {d^6}{a^4b}\right )^{-3}$	From previous work.




	$= \left (\frac {b^{10}c^5d^{45}}{a^{15}}\right ) \left (\frac {d^{-18}}{a^{-12}b^{-3}}\right )$	Distribute Power.




	$= \frac {b^{10}c^5d^{45}d^{-18}}{a^{15}a^{-12}b^{-3}}$	Multiple Straight Across.




	$= \frac {b^{13}c^5d^{27}}{a^3}$	Simplify powers using Exponential Properties.




	$= a^{-3}b^{13}c^5d^{27}$	Rewrite using negative exponents to avoid having a fraction.

1 : Use rules of exponents to rewrite the following expression without any fractions (using negative exponents if needed). $\left (\frac {\sage {p1f1}}{\sage {p1f2}}\right )^{\sage {p1pwr1}} \cdot \left (\frac {\sage {p1f3}}{\sage {p1f4}}\right )^{\sage {p1pwr2}} = x^{\answer {\sage {p1ans1}}}\cdot y^{\answer {\sage {p1ans2}}}\cdot z^{\answer {\sage {p1ans3}}} \cdot r^{\answer {\sage {p1ans4}}}$

Follow the walkthrough above closely; start with simplifying the inside of each set of parentheses, then distribute the power from the parentheses, then combine the two results into a single fraction.

2 : Use rules of exponents to rewrite the following expression without any fractions (using negative exponents if needed). $\left (\frac {\sage {p2f1}}{\sage {p2f2}}\right )^{\sage {p2pwr1}} \cdot \left (\frac {\sage {p2f3}}{\sage {p2f4}}\right )^{\sage {p2pwr2}} = x^{\answer {\sage {p2ans1}}}\cdot y^{\answer {\sage {p2ans2}}}\cdot z^{\answer {\sage {p2ans3}}} \cdot r^{\answer {\sage {p2ans4}}}$

3 : Use rules of exponents to rewrite the following expression without any fractions (using negative exponents if needed). $\left (\frac {\sage {p3f1}}{\sage {p3f2}}\right )^{\sage {p3pwr1}} \cdot \left (\frac {\sage {p3f3}}{\sage {p3f4}}\right )^{\sage {p3pwr2}} = x^{\answer {\sage {p3ans1}}}\cdot y^{\answer {\sage {p3ans2}}}\cdot z^{\answer {\sage {p3ans3}}} \cdot r^{\answer {\sage {p3ans4}}}$

4 : Use rules of exponents to rewrite the following expression without any fractions (using negative exponents if needed). $\left (\frac {\sage {p4f1}}{\sage {p4f2}}\right )^{\sage {p4pwr1}} \cdot \left (\frac {\sage {p4f3}}{\sage {p4f4}}\right )^{\sage {p4pwr2}} = x^{\answer {\sage {p4ans1}}}\cdot y^{\answer {\sage {p4ans2}}}\cdot z^{\answer {\sage {p4ans3}}} \cdot r^{\answer {\sage {p4ans4}}}$

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