Given \( x>0 \) and \( y>0 \), select the expression that is equivalent to \[ \sqrt[4]{81 x^{8} y^{18}} \] Answer \( 9 x^{2} y^{\frac{9}{2}} \) \( 9 x^{\frac{1}{2}} y^{\frac{2}{9}} \)

Calculate or simplify the expression \( \sqrt[4]{81x^8y^{18}} \). Simplify the expression by following steps: - step0: Solution: \(\sqrt[4]{81x^{8}y^{18}}\) - step1: Transform the expression: \(\sqrt[4]{\left(9x^{4}y^{9}\right)^{2}}\) - step2: Calculate: \(\sqrt{9x^{4}y^{9}}\) - step3: Rewrite the exponent: \(\sqrt{9x^{4}y^{8+1}}\) - step4: Expand the expression: \(\sqrt{9x^{4}y^{8}\times y}\) - step5: Reorder the terms: \(\sqrt{x^{4}y^{8}\times 9y}\) - step6: Use the properties of radicals: \(\sqrt{x^{4}y^{8}}\times \sqrt{9y}\) - step7: Simplify the root: \(y^{4}x^{2}\sqrt{9y}\) The expression equivalent to \(\sqrt[4]{81x^{8}y^{18}}\) is \(y^{4}x^{2}\sqrt{9y}\). Therefore, the correct answer is \(9x^{2}y^{\frac{9}{2}}\).