((x^2/4)^3)/((x^3))^{9/4= square (Simplify your answer. Type exponential notation

To simplify the expression \(\frac{(x^{2/4})^3}{(x^3)^{9/4}}\), we can use the properties of exponents.<br /><br />First, let's simplify the numerator \((x^{2/4})^3\):<br />\[<br />(x^{2/4})^3 = (x^{1/2})^3 = x^{3/2}<br />\]<br /><br />Next, let's simplify the denominator \((x^3)^{9/4}\):<br />\[<br />(x^3)^{9/4} = x^{27/4}<br />\]<br /><br />Now, we can rewrite the original expression using these simplified forms:<br />\[<br />\frac{x^{3/2}}{x^{27/4}}<br />\]<br /><br />To divide exponents with the same base, we subtract the exponent in the denominator from the exponent in the numerator:<br />\[<br />x^{3/2 - 27/4}<br />\]<br /><br />To perform the subtraction, we need a common denominator. The common denominator for 2 and 4 is 4. So, we convert \(3/2\) to a fraction with a denominator of 4:<br />\[<br />3/2 = 6/4<br />\]<br /><br />Now, we can subtract the exponents:<br />\[<br />6/4 - 27/4 = -21/4<br />\]<br /><br />Therefore, the simplified expression is:<br />\[<br />x^{-21/4}<br />\]<br /><br />So, the final answer is:<br />\[<br />\boxed{x^{-21/4}}<br />\]