Question

1. \\((3x^{-2}y^{3})^{2}\\)

Explanation:

Step1: Apply power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). So, we can apply this to \((3x^{-2}y^{3})^{2}\) as follows:
\((3x^{-2}y^{3})^{2}=3^{2}\times(x^{-2})^{2}\times(y^{3})^{2}\)

Step2: Simplify each term

• For \(3^{2}\), we know that \(3\times3 = 9\).
• For \((x^{-2})^{2}\), we use the power of a power rule \((a^{m})^{n}=a^{m\times n}\). So, \((x^{-2})^{2}=x^{-2\times2}=x^{-4}\). We can also rewrite \(x^{-4}\) as \(\frac{1}{x^{4}}\) using the negative exponent rule \(a^{-n}=\frac{1}{a^{n}}\), but we can keep it as \(x^{-4}\) for now and combine later or simplify further.
• For \((y^{3})^{2}\), using the power of a power rule, we get \(y^{3\times2}=y^{6}\).

Now, combining these results:
\(3^{2}\times(x^{-2})^{2}\times(y^{3})^{2}=9\times x^{-4}\times y^{6}\)
We can rewrite \(x^{-4}\) as \(\frac{1}{x^{4}}\), so the expression becomes \(\frac{9y^{6}}{x^{4}}\) or \(9x^{-4}y^{6}\).

Answer:

\(9x^{-4}y^{6}\) (or \(\frac{9y^{6}}{x^{4}}\))