To express \( 8\sqrt{7776} \) with a radical exponent, we can rewrite the radical as a fractional exponent.
The expression \( \sqrt{7776} \) is equivalent to \( 7776^{1/2} \). Thus, we can write:
\[ 8\sqrt{7776} = 8 \cdot 7776^{1/2} \]
Next, we can express the number 8 as \( 8^{1} \), and then rewrite the entire expression as:
\[ 8^{1} \cdot 7776^{1/2} \]
If we want to combine these using exponent rules, we also note that \( 8^{1} \) can be expressed as \( (2^3)^1 = 2^3 \). Therefore, we can write:
\[ 8\sqrt{7776} = 2^3 \cdot 7776^{1/2} \]
To find the actual value of \( 8\sqrt{7776} \), first, let's simplify \( \sqrt{7776} \).
We start by factorizing 7776:
\[ 7776 = 2^5 \times 3^5 \]
Now, we can find the square root:
\[ \sqrt{7776} = \sqrt{2^5 \times 3^5} = \sqrt{2^4 \times 2^1} \times \sqrt{3^4 \times 3^1} = 2^2 \times 3^2 \times \sqrt{2 \cdot 3} \] \[ = 4 \times 9 \times \sqrt{6} = 36\sqrt{6} \]
Now substituting this back into the expression, we get:
\[ 8\sqrt{7776} = 8 \cdot 36\sqrt{6} = 288\sqrt{6} \]
Thus, the radical exponent representation of \( 8\sqrt{7776} \) is \( 8 \cdot 7776^{1/2} \), and the numerical value is:
\[ \boxed{288\sqrt{6}} \]