1. **State the problem:** Rewrite the expression $\left(27 \sqrt[3]{z^2}\right)^{\frac{1}{3}}$ in the form $k \cdot z^n$ where $n$ is an integer, fraction, or exact decimal. 2. **Recall the rules:** - $a^{m/n} = \sqrt[n]{a^m}$. - When raising a product to a power, apply the power to each factor: $(ab)^c = a^c b^c$. - When raising a power to another power, multiply exponents: $(a^m)^n = a^{mn}$. 3. **Rewrite the expression:** $$\left(27 \sqrt[3]{z^2}\right)^{\frac{1}{3}} = \left(27 \cdot z^{\frac{2}{3}}\right)^{\frac{1}{3}}$$ 4. **Apply the power to each factor:** $$= 27^{\frac{1}{3}} \cdot \left(z^{\frac{2}{3}}\right)^{\frac{1}{3}}$$ 5. **Simplify each term:** - $27^{\frac{1}{3}} = 3$ because $3^3 = 27$. - For the $z$ term, multiply exponents: $\frac{2}{3} \times \frac{1}{3} = \frac{2}{9}$. 6. **Final expression:** $$3 \cdot z^{\frac{2}{9}}$$ Thus, the expression in the form $k \cdot z^n$ is $3 \cdot z^{\frac{2}{9}}$.