To express \( 5\sqrt{7776} \) using rational exponents, we first note that the square root can be written as an exponent of \( \frac{1}{2} \). Therefore, we can rewrite \( \sqrt{7776} \) as \( 7776^{\frac{1}{2}} \).
Thus, we have:
\[ 5\sqrt{7776} = 5 \cdot 7776^{\frac{1}{2}} \]
If you want it in a single expression, you can also express this as:
\[ 5\sqrt{7776} = 5 \cdot 7776^{\frac{1}{2}} = 5^{1} \cdot 7776^{\frac{1}{2}} \]
Both forms are correct, but the first one clearly indicates the usage of a rational exponent.