(27sqrt [3](z^2))^(1)/(3)=square

To simplify the expression \((27\sqrt[3]{z^2})^{\frac{1}{3}}\), we can follow these steps:<br /><br />1. **Rewrite the expression inside the parentheses:**<br /> \[<br /> 27\sqrt[3]{z^2} = 27 \cdot z^{\frac{2}{3}}<br /> \]<br /><br />2. **Apply the exponent \(\frac{1}{3}\) to the entire expression:**<br /> \[<br /> \left(27 \cdot z^{\frac{2}{3}}\right)^{\frac{1}{3}}<br /> \]<br /><br />3. **Distribute the exponent \(\frac{1}{3}\) to both terms inside the parentheses:**<br /> \[<br /> 27^{\frac{1}{3}} \cdot \left(z^{\frac{2}{3}}\right)^{\frac{1}{3}}<br /> \]<br /><br />4. **Simplify each term separately:**<br /> - For \(27^{\frac{1}{3}}\):<br /> \[<br /> 27^{\frac{1}{3}} = \sqrt[3]{27} = 3<br /> \]<br /> - For \(\left(z^{\frac{2}{3}}\right)^{\frac{1}{3}}\):<br /> \[<br /> \left(z^{\frac{2}{3}}\right)^{\frac{1}{3}} = z^{\left(\frac{2}{3} \cdot \frac{1}{3}\right)} = z^{\frac{2}{9}}<br /> \]<br /><br />5. **Combine the simplified terms:**<br /> \[<br /> 3 \cdot z^{\frac{2}{9}}<br /> \]<br /><br />Therefore, the simplified form of the given expression is:<br />\[<br />\boxed{3z^{\frac{2}{9}}}<br />\]