What values of \( c \) and \( d \) make the equation true? Assume \( x>0 \) and \( y \geq 0 \) \( \sqrt{\frac{50 x^{6} y^{3}}{9 x^{8}}}=\frac{5 y^{6} \sqrt{2 y}}{d x} \)

To solve the equation \( \sqrt{\frac{50 x^{6} y^{3}}{9 x^{8}}}=\frac{5 y^{6} \sqrt{2 y}}{d x} \), we start by simplifying the left-hand side. 1. Simplify the left side: \[ \sqrt{\frac{50 x^{6} y^{3}}{9 x^{8}}} = \sqrt{\frac{50 y^{3}}{9 x^{2}}} = \frac{\sqrt{50 y^{3}}}{3 x} \] Next, we calculate \( \sqrt{50 y^3} \): \[ \sqrt{50 y^{3}} = \sqrt{25 \cdot 2 \cdot y^{3}} = 5 y^{3/2} \sqrt{2} \] So, the left-hand side becomes: \[ \frac{5 y^{3/2} \sqrt{2}}{3 x} \] 2. Set the left-hand side equal to the right-hand side: \[ \frac{5 y^{3/2} \sqrt{2}}{3 x} = \frac{5 y^{6} \sqrt{2 y}}{d x} \] 3. We can divide both sides by \( \sqrt{2} x \) (since \( x > 0 \)): \[ \frac{5 y^{3/2}}{3} = \frac{5 y^{6} \sqrt{y}}{d} \] 4. Canceling \( 5 \) from both sides gives: \[ \frac{y^{3/2}}{3} = \frac{y^{6} \sqrt{y}}{d} \] We can express \( \sqrt{y} \) as \( y^{1/2} \), so the right-hand side simplifies to: \[ \frac{y^{6} y^{1/2}}{d} = \frac{y^{6.5}}{d} = \frac{y^{13/2}}{d} \] This means we now have: \[ \frac{y^{3/2}}{3} = \frac{y^{13/2}}{d} \] 5. Next, multiply both sides by \( d \) to clear the fraction: \[ d \cdot \frac{y^{3/2}}{3} = y^{13/2} \] 6. You can now divide both sides by \( y^{3/2} \) (valid since \( y \geq 0 \) and we assume \( y \neq 0 \)): \[ \frac{d}{3} = y^{13/2 - 3/2} = y^{10/2} = y^{5} \] 7. To find the value of \( d \), we set \( y \) to the appropriate power, but since we want this equation to hold for all \( y \geq 0 \), it should be independent of \( y \). Thus, we need the coefficient of \( y^5 \) from the left side to match a specific value. We observe that for \( y^{5} \) to balance across the equation, we need: \[ d = 3 \] Now, substitute back into the equation to find \( c \): From step 6, if \( d = 3 \), we can take any value of \( c \). By the structure, it confirms \( c = 1 \) would work. Thus, \( c = 1 \) and \( d = 3 \).