Let's solve the given expression step by step: Given: \[ y = \frac{8x^2 + 8x + 4}{\sqrt{x}} \] ### Step 1: Split the Fraction Break the numerator into separate terms over the denominator: \[ y = \frac{8x^2}{\sqrt{x}} + \frac{8x}{\sqrt{x}} + \frac{4}{\sqrt{x}} \] ### Step 2: Simplify Each Term Recall that \(\sqrt{x} = x^{1/2}\). - \(\frac{8x^2}{x^{1/2}} = 8x^{2 - 1/2} = 8x^{3/2}\) - \(\frac{8x}{x^{1/2}} = 8x^{1 - 1/2} = 8x^{1/2}\) - \(\frac{4}{x^{1/2}} = 4x^{-1/2}\) ### Step 3: Combine the Terms \[ y = 8x^{3/2} + 8x^{1/2} + 4x^{-1/2} \] ### **Final Answer** \[ \boxed{y = 8x^{3/2} + 8x^{1/2} + 4x^{-1/2}} \] This is the simplified form of the given expression.

The expression is simplified by dividing each term in the numerator by the square root of \(x\). This reduces exponents using the rule \(x^a / x^b = x^{a-b}\), resulting in terms with exponents \(3/2\), \(1/2\), and \(-1/2\).