Question

1. ((x^2 y^{-1})^2)
2. ((3m)^{-2})
3. (\frac{x^{-1}}{4x^4})

Explanation:

Problem 18: \(\boldsymbol{(x^2 y^{-1})^2}\)

Step1: Apply power of a product rule

\((ab)^n = a^n b^n\), so \((x^2 y^{-1})^2=(x^2)^2\cdot(y^{-1})^2\)

Step2: Apply power of a power rule

\((a^m)^n = a^{mn}\), so \((x^2)^2 = x^{2\times2}=x^4\) and \((y^{-1})^2 = y^{-1\times2}=y^{-2}\)

Step3: Rewrite negative exponent

\(a^{-n}=\frac{1}{a^n}\), so \(y^{-2}=\frac{1}{y^2}\)
Combining, we get \(x^4\cdot y^{-2}=\frac{x^4}{y^2}\)

Step1: Apply power of a product rule

\((ab)^n = a^n b^n\), so \((3m)^{-2}=3^{-2}\cdot m^{-2}\)

Step2: Rewrite negative exponents

\(a^{-n}=\frac{1}{a^n}\), so \(3^{-2}=\frac{1}{3^2}=\frac{1}{9}\) and \(m^{-2}=\frac{1}{m^2}\)
Combining, we get \(\frac{1}{9}\cdot\frac{1}{m^2}=\frac{1}{9m^2}\)

Step1: Apply quotient rule for exponents

\(\frac{a^m}{a^n}=a^{m - n}\), so \(\frac{x^{-1}}{x^4}=x^{-1-4}=x^{-5}\)

Step2: Rewrite negative exponent

\(a^{-n}=\frac{1}{a^n}\), so \(x^{-5}=\frac{1}{x^5}\)
Thus, \(\frac{x^{-1}}{4x^4}=\frac{1}{4x^5}\)

Answer:

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

Problem 20: \(\boldsymbol{(3m)^{-2}}\)