Question

1. $(2x^{4}y^{-3})^{-1}$

Explanation:

Step1: Apply the power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). So, for \((2x^{4}y^{-3})^{-1}\), we can apply this rule to each factor inside the parentheses:
\(2^{-1}(x^{4})^{-1}(y^{-3})^{-1}\)

Step2: Apply the power of a power rule

The power of a power rule states that \((a^m)^n = a^{mn}\). Let's apply this to each term:

• For \(2^{-1}\), it remains as \(2^{-1}\).
• For \((x^{4})^{-1}\), we have \(x^{4\times(-1)} = x^{-4}\).
• For \((y^{-3})^{-1}\), we have \(y^{-3\times(-1)} = y^{3}\).

So now we have \(2^{-1}x^{-4}y^{3}\).

Step3: Rewrite negative exponents

Recall that \(a^{-n}=\frac{1}{a^{n}}\). So, \(2^{-1}=\frac{1}{2}\) and \(x^{-4}=\frac{1}{x^{4}}\).
Substituting these back in, we get \(\frac{1}{2}\times\frac{1}{x^{4}}\times y^{3}=\frac{y^{3}}{2x^{4}}\).

Answer:

\(\frac{y^{3}}{2x^{4}}\)