Final answer:
To find the values of c and d that satisfy the equations, we compare the exponents and coefficients on both sides of Equation A and B. By equating these terms, we find that for Equation A, c must be 6, and for Equation B, d must be 9.
Step-by-step explanation:
If x>0, the values of c and d that make the equations true are determined by simplifying each side of Equations A and B to match the corresponding terms. Starting with Equation A: \(\sqrt{448x^c} = 8x^3\sqrt{7x}\), we simplify the left hand side by factoring 448 into \(64 \times 7\), and recognizing that \(\sqrt{64}=8\) and \(\sqrt{7}\) can be taken out of the square root, leaving us with \(8x^{\frac{c}{2}}\sqrt{7x} = 8x^3\sqrt{7x}\). By equating the exponents of x on both sides of the equation, we find that \(\frac{c}{2} = 3\), which suggests c = 6. Next, we look at Equation B: \(^3\sqrt{576x^d} = 4x^3\sqrt{9x^2}\). The cube root of 576 is 24, but we can simplify this to \(4 \times 6\), knowing that 4 is already on the right-hand side and recognizing that \(^3\sqrt{6}\) can be represented as \(6^{\frac{1}{3}}\). Thus, we have \(4x^{\frac{d}{3}}\sqrt{x^2} = 4x^3\sqrt{9x^2}\), and by equating the apropriate terms, we determine that \(\frac{d}{3}=3\), so d = 9.