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}
