Question

1. \\(\frac{(2x^{-1}y^{0})^{4}}{2xy^{-1}}\\)

Explanation:

Step1: Apply Power of a Product Rule

For \((2x^{-1}y^{0})^{4}\), use \((ab)^n = a^n b^n\) and \((a^m)^n = a^{mn}\), \(y^0 = 1\). So:
\((2)^4(x^{-1})^4(y^0)^4 = 16x^{-4}(1) = 16x^{-4}\)

Step2: Rewrite the Fraction

Now the expression is \(\frac{16x^{-4}}{2xy^{-1}}\). Separate coefficients and variables:
\(\frac{16}{2} \cdot \frac{x^{-4}}{x} \cdot \frac{1}{y^{-1}}\)

Step3: Simplify Coefficients and Exponents

Simplify coefficients: \(\frac{16}{2}=8\). For variables, use \(a^m / a^n = a^{m - n}\) and \(1/a^{-n}=a^n\):
\(\frac{x^{-4}}{x}=x^{-4 - 1}=x^{-5}\), \(\frac{1}{y^{-1}} = y^{1}\)

Step4: Combine Results

Multiply together: \(8x^{-5}y = \frac{8y}{x^5}\)

Answer:

\(\frac{8y}{x^5}\)