54. \( \frac{4 \sqrt{5}}{3 \sqrt{8}} \)

To simplify \( \frac{4 \sqrt{5}}{3 \sqrt{8}} \), you first need to simplify the denominator. Since \( \sqrt{8} = \sqrt{4 \cdot 2} = 2 \sqrt{2} \), we can rewrite the expression as: \[ \frac{4 \sqrt{5}}{3 \cdot 2 \sqrt{2}} = \frac{4 \sqrt{5}}{6 \sqrt{2}} = \frac{2 \sqrt{5}}{3 \sqrt{2}} \] Next, we can rationalize the denominator by multiplying the numerator and denominator by \( \sqrt{2} \): \[ \frac{2 \sqrt{5} \cdot \sqrt{2}}{3 \sqrt{2} \cdot \sqrt{2}} = \frac{2 \sqrt{10}}{3 \cdot 2} = \frac{\sqrt{10}}{3} \] So, \( \frac{4 \sqrt{5}}{3 \sqrt{8}} = \frac{\sqrt{10}}{3} \).