Kensa is simplifying the following problem and justifying her steps. Help her fill in the blanks where she is missing information.

[tex]\[
\begin{tabular}{|c|l|}
\hline
$x^{-3} y^6 \cdot (3 x^2 y)^{-2}$ & Original/Given \\
\hline
\frac{y^6}{x^3} \cdot \left(\frac{1}{3 x^2 y}\right)^2 & Rewriting the negative exponents to become positive \\
\hline
\frac{y^6}{x^3} \cdot \frac{1}{9 x^4 y^2} & a) Power of a product \\
\hline
\frac{y^6}{9 x^7 y^4} & b) Simplifying the expression \\
\hline
\frac{y^2}{9 x^7} & c) Simplifying the fraction \\
\hline
\end{tabular}
\][/tex]



Answer :

Let's provide the detailed, step-by-step solution so Kensa can understand each step in the process.

1. Original Expression:
[tex]\[ x^{-3} y^6 \cdot \left(3 x^2 y\right)^{-2} \][/tex]
This is the given expression.

2. Rewrite Negative Exponents:
[tex]\[ \frac{y^6}{x^3} \cdot \left(\frac{1}{3 x^2 y^2}\right)^2 \][/tex]
Here, we rewrote the negative exponents. [tex]\( x^{-3} \)[/tex] was rewritten as [tex]\( \frac{1}{x^3} \)[/tex] and [tex]\( (3 x^2 y)^{-2} \)[/tex] was rewritten as [tex]\( \left(\frac{1}{3 x^2 y^2}\right)^2 \)[/tex].

3. Apply the Power of a Product Rule:
When applying the power rule [tex]\(\left(\frac{1}{3 x^2 y^2}\right)^2\)[/tex], we get:
[tex]\[ \left(\frac{1}{3 x^2 y^2}\right)^2 = \frac{1}{(3 x^2 y^2)^2} = \frac{1}{9 x^4 y^4} \][/tex]
So, the expression becomes:
[tex]\[ \frac{y^6}{x^3} \cdot \frac{1}{9 x^4 y^4} \][/tex]
Thus, for the missing part 'a)', we have:
a) [tex]\(\frac{1}{9 x^4 y^4}\)[/tex]

4. Combine the Results:
Now, we combine the two fractions:
[tex]\[ \frac{y^6}{x^3} \cdot \frac{1}{9 x^4 y^4} = \frac{y^6 \cdot 1}{x^3 \cdot 9 x^4 y^4} = \frac{y^6}{9 x^7 y^4} \][/tex]
So, the step b) should be:
b) [tex]\[ \frac{y^6}{9 x^7 y^4} \][/tex]

5. Simplify the Expression:
Finally, we simplify the expression by subtracting the exponents for [tex]\( y \)[/tex]:
[tex]\[ \frac{y^6}{9 x^7 y^4} = \frac{y^{6-4}}{9 x^7} = \frac{y^2}{9 x^7} \][/tex]

So for the final missing part c), we have:
c) [tex]\(\frac{y^2}{9 x^7}\)[/tex]

Here’s the completed table with all the steps filled in:

\begin{tabular}{|c|l|}
\hline[tex]$x^{-3} y^6 \cdot\left(3 x^2 y\right)^{-2}$[/tex] & Original/Given \\
\hline[tex]$\frac{y^6}{x^3} \cdot\left(\frac{1}{3 x^2 y^2}\right)^2$[/tex] & \begin{tabular}{l}
Rewriting the negative exponents \\
to become positive.
\end{tabular} \\
\hline[tex]$\frac{1}{9 x^4 y^4}$[/tex] & Power of a product \\
\hline[tex]$\frac{y^6}{9 x^7 y^4}$[/tex] & Combine the results \\
\hline[tex]$\frac{y^2}{9 x^7}$[/tex] & Simplify the expression \\
\hline
\end{tabular}